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BEGIN:VEVENT
SUMMARY:Tanmay Deshpande (TIFR)
DTSTART:20200408T203000Z
DTEND:20200408T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/1
DESCRIPTION:Title: Character sheaves on algebraic groups\nby Tanmay Deshpande (TIFR
) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\
nCharacter sheaves on an algebraic group are supposed to be the geometric
analogues of irreducible characters of a finite group. In 1980s Lusztig de
veloped the\ntheory of character sheaves on reductive groups and gave a ge
ometric description\nof the character theory of finite reductive groups. I
nspired by Lusztig’s works\,\nBoyarchenko and Drinfeld developed the the
ory of character sheaves on unipotent\ngroups. In this talk\, I will descr
ibe an approach (due to Drinfeld) towards a theory\nof character sheaves o
n general algebraic groups and describe the known results in\nthe case of
solvable algebraic groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Mautner (Dartmouth)
DTSTART:20200415T203000Z
DTEND:20200415T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/2
DESCRIPTION:Title: Shadows of Lie theory in the world of matroids\nby Carl Mautner
(Dartmouth) as part of MIT Lie groups seminar\n\n\nAbstract\nI will discus
s a program (partly conjectural) exploring analogues of the Schur\nalgebra
and category $\\mathcal O$ for matroids and oriented matroids. This progr
am was motivated in large part by work of Braden-Licata-Proudfoot-Webster.
The talk\nwill be based on joint work with Tom Braden and work in progres
s with Jens\nEberhardt and Ethan Kowalenko. I will not assume prior knowle
dge of matroid\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clifton Cunningham (University of Calgary)
DTSTART:20200506T203000Z
DTEND:20200506T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/3
DESCRIPTION:Title: Arthur packets for G(2) and perverse sheaves on cubics\nby Clift
on Cunningham (University of Calgary) as part of MIT Lie groups seminar\n\
n\nAbstract\nThis talk demonstrates a non-invasive procedure that calculat
es Arthur packets\, their associated stable distributions and Langlands-Sh
elstad transfers\, without direct use of endoscopy\, using certain unipote
nt representations of the split p-adic exceptional group G(2) as examples.
In the case at hand\, this procedure relies on a study of the category of
GL(2)-equivariant perverse sheaves on the moduli space of homogeneous cub
ics in two variables\, which is perhaps of independent interest. Specifica
lly\, we find the Fourier transform and the microlocalization of the simpl
e objects in this category\, and convert that into information about the A
ubert involution and stable distributions attached to Arthur packets. This
is joint work with Andrew Fiori and Qing Zhang\, based on earlier joint w
ork with Andrew Fiori\, Ahmed Moussaoui\, James Mracek and Bin Xu\, which
is based on earlier work by David Vogan\, sadly\, not joint.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucas Mason-Brown (MIT)
DTSTART:20200429T203000Z
DTEND:20200429T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/4
DESCRIPTION:Title: Unipotent representations of real reductive groups\nby Lucas Mas
on-Brown (MIT) as part of MIT Lie groups seminar\n\n\nAbstract\nLet $G$ be
a real reductive group and let ${\\widehat G}$ be the set of\nirreducible
unitary representations of $G$. The determination of $\\widehat G$ (for\n
arbitrary $G$) is one of the fundamental unsolved problems in\nrepresentat
ion theory. In the early 1980s\, Arthur introduced a finite\nset Unip($G$)
of (conjecturally unitary) irreducible representations of\n$G$ called {\\
it unipotent representations}. In a certain sense\, these\nrepresentations
form the building blocks of $\\widehat G$. Hence\, the\ndetermination of
$\\widehat G$ requires as a crucial ingredient the determination\nof Unip(
$G$). In this thesis\, we prove three results on unipotent\nrepresentation
s. First\, we study unipotent representations by\nrestriction to $K\\subs
et G$\, a maximal compact subgroup. We deduce a formula\nfor this restrict
ion in a wide range of cases\, proving (in these\ncases) a long-standing c
onjecture of Vogan. Next\, we study the\nunipotent representations attache
d to induced nilpotent orbits. We\nfind that Unip($G$) is ‘generated’
by an even smaller set $\\hbox{Unip}'(G)$\nconsisting of representations a
ttached to rigid nilpotent\norbits. Finally\, we study the unipotent repre
sentations attached to\nthe principal nilpotent orbit. We provide a comple
te classification of\nsuch representations\, including a formula for their
$K$-types.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Bezrukavnikov (MIT)
DTSTART:20200422T203000Z
DTEND:20200422T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/5
DESCRIPTION:Title: Canonical bases and coherent sheaves\nby Roman Bezrukavnikov (MI
T) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract
\nThe primary application of canonical bases in a Grothendieck group of\nr
epresentations is to computation of characters of (say) irreducible\nrepre
sentations\; however\, it is not their only application. I will\nreview c
onstruction and properties of canonical bases in Grothendieck\ngroups of c
oherent sheaves on the Springer resolution and related\nspaces and specula
te on possible generalization to a new setting\ninvolving the fixed group
of an involution. The toolbox includes\nlinear Koszul duality of Mirkovic-
Riche and a version of Soergel\nbimodules theory.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20200513T203000Z
DTEND:20200513T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/6
DESCRIPTION:Title: Unipotent representations\nby Jeffrey Adams (University of Maryl
and) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstra
ct\nI will give an overview of the current state of the Atlas of Lie group
s and Representations project\, with an emphasis on computing all unipoten
t representations\nof real exceptional groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT Mathematics)
DTSTART:20200909T203000Z
DTEND:20200909T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/7
DESCRIPTION:Title: Structure of Harish-Chandra cells\nby David Vogan (MIT Mathemati
cs) as part of MIT Lie groups seminar\n\n\nAbstract\nOne of the fundamenta
l contributions of Kazhdan and Lusztig's 1979 Inventiones paper was the no
tion of "cells" in Weyl groups. They gave a decomposition of the left regu
lar representation of W as a direct sum of "left cell" representations\, w
hich encode deep and powerful information about group representations. In
the case of the symmetric group S_n=W\, the left cells are irreducible rep
resentations. In all other cases they are not. Lusztig in his 1984 book ga
ve a beautiful description of all left cells in terms of the geometry of a
nilpotent orbit.\n\\\\\nThere is a parallel notion of "Harish-Chandra cel
ls" in the representation theory of a real reductive group G(R). Again eac
h cell is a representation of W\, encoding deep information about the G(R)
representations. I will formulate a conjecture extending Lusztig's calcul
ation of left cell representations to this case\, and explain its connecti
on with Arthur's theory of unipotent representations.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Chan (MIT Mathematics)
DTSTART:20200916T203000Z
DTEND:20200916T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/8
DESCRIPTION:Title: A strong Henniart identity for reductive groups over finite fields\nby Charlotte Chan (MIT Mathematics) as part of MIT Lie groups seminar\
n\nLecture held in 2-142.\n\nAbstract\nIn 1992\, Henniart proved that supe
rcuspidal representations for –adic GLn are determined by their characte
r on so-called very regular elements. This has been useful in many ways
as it allows for convenient comparison between various constructions of su
percuspidal representations for GLn. We describe a version of this type
of result which holds for (some) representations of reductive groups over
finite fields. This is joint work with Masao Oi.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Wang (MIT Mathematics)
DTSTART:20200923T203000Z
DTEND:20200923T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/9
DESCRIPTION:Title: Spherical varieties\, L-functions\, and crystal bases\nby Jonath
an Wang (MIT Mathematics) as part of MIT Lie groups seminar\n\nLecture hel
d in 2-142.\n\nAbstract\nThe program of Sakellaridis and Venkatesh propose
s a unified framework to study integral representations of L-functions thr
ough the lens of spherical varieties. For X an affine spherical variety\,
the (hypothetical) IC complex of the infinite-dimensional formal arc space
of X is conjecturally related to special values of local unramified L-fun
ctions. We formulate this relation precisely using a new conjectural geome
tric construction of the crystal basis of a finite-dimensional representat
ion (determined by X) of the dual group. We prove these conjectures for a
large class of spherical varieties. This is joint work with Yiannis Sakell
aridis.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Dudas (CNRS)
DTSTART:20200930T203000Z
DTEND:20200930T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/10
DESCRIPTION:Title: Macdonald polynomials and decomposition numbers for finite unitary
groups\nby Olivier Dudas (CNRS) as part of MIT Lie groups seminar\n\nL
ecture held in 2-142.\n\nAbstract\n(work in progress with R. Rouquier) In
this talk I will present a computational (yet conjectural) method to deter
mine some decomposition matrices for finite groups of Lie type. I will fir
st explain how one can produce a "natural" self-equivalence in the case of
$\\mathrm{GL}_n(q)$ coming from the topology of the Hilbert scheme of $\\
mathbb{C}^2$. The combinatorial part of this equivalence is related to Mac
donald's theory of symmetric functions and gives $(q\,t)$-decomposition nu
mbers. The evidence suggests that the case of finite unitary groups is obt
ained by taking a suitable square root of that equivalence.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Ostrik (University of Oregon)
DTSTART:20201007T203000Z
DTEND:20201007T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/11
DESCRIPTION:Title: Two Dimensional Field Theories and Partial Fractions\nby Victor
Ostrik (University of Oregon) as part of MIT Lie groups seminar\n\nLectur
e held in 2-142.\n\nAbstract\nThis talk is based on joint work with M.Khov
anov and Y.Kononov. By evaluating a topological field theory in dimension
2 on surfaces of genus 0\,1\,2 etc we get a sequence. We investigate which
sequences occur in this way depending on the assumptions on the target ca
tegory.\n\n\n\n\n\nPlease become a member of our email list to receive ann
ouncements of upcoming MIT Lie Groups seminars as well as related informat
ion:\n\nhttps://mailman.mit.edu:444/mailman/listinfo/liegroups\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maarten Solleveld (Radboud Universiteit)
DTSTART:20201014T203000Z
DTEND:20201014T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/12
DESCRIPTION:Title: Bernstein components for p-adic groups\nby Maarten Solleveld (R
adboud Universiteit) as part of MIT Lie groups seminar\n\nLecture held in
2-142.\n\nAbstract\nSuppose that one has a supercuspidal representation of
a Levi subgroup of some reductive $p$-adic group $G$. Bernstein associate
d to this a block Rep$(G)^s$ in the category of smooth $G$-representations
. We address the question: what does Rep$(G)^s$ look like?\n\nUsually this
is investigated with Bushnell--Kutzko types\, but these are not always av
ailable. Instead\, we approach it via the endomorphism algebra of a progen
erator of Rep$(G)^s$. We will show that Rep$(G)^s$ is "almost" equivalent
with the module category of an affine Hecke algebra -- a statement that wi
ll be made precise in several ways.\n\nIn the end\, this leads to a classi
fication of the irreducible representations in Rep$(G)^s$ in terms of the
complex torus and the finite groups that are canonically associated to thi
s Bernstein component.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dima Arinkin (University of Wisconsin)
DTSTART:20201021T203000Z
DTEND:20201021T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/13
DESCRIPTION:Title: Compactifying the category of D-modules on the stack of G-bundles\nby Dima Arinkin (University of Wisconsin) as part of MIT Lie groups se
minar\n\nLecture held in 2-142.\n\nAbstract\nLet X be a projective curve\,
G a reductive group. Let Bun be the stack of G-bundles over X\, and consi
der the category of D-modules on Bun. (This category appears on the “aut
omorphic” side of the geometric Langlands correspondence.) Drinfeld and
Gaitsgory prove that\, despite the “unbounded” (non-quasi compact) nat
ure of Bun\, the category of D-modules is well-behaved (compactly generate
d).\n\nIn this talk\, we will “compactify” this category in a stronger
sense\; this can be viewed as compactifying the quantized cotangent bundl
e to Bun. While the basic idea of such compactification goes back to ideas
of Deligne and Simpson\, its construction relies on non-trivial propertie
s of the geometry of Bun (similar to the Drinfeld-Gaitsgory Theorem).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joel Kamnitzer (University of Toronto)
DTSTART:20201028T203000Z
DTEND:20201028T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/14
DESCRIPTION:Title: Categorical g-actions for modules over truncated shifted Yangians\nby Joel Kamnitzer (University of Toronto) as part of MIT Lie groups se
minar\n\nLecture held in 2-142.\n\nAbstract\nGiven a representation V of a
reductive group G\, Braverman-Finkelberg-Nakajima defined a Poisson varie
ty called the Coulomb branch\, using a convolution algebra construction. T
his variety comes with a natural deformation quantization\, called a Coulo
mb branch algebra. Important cases of these Coulomb branches are (generali
zed) affine Grassmannian slices\, and their quantizations are truncated sh
ifted Yangians.\n\nMotivated by the geometric Satake correspondence and th
e theory of symplectic duality/3d mirror symmetry\, we expect a categorica
l g-action on modules for these truncated shifted Yangians. I will explain
three results in this direction. First\, we have an indirect realization
of this action\, using equivalences with KLRW-modules. Second\, we have a
geometric relation between these generalized slices by Hamiltonian reducti
on. Finally\, we have an algebraic version of this Hamiltonian reduction w
hich we are able to relate to the first realization.\n\nThis seminar will
take place entirely online. Please email Andre Dixon (aldixon@mit.edu) for
the Zoom meeting Link.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harrison Chen (Cornell University)
DTSTART:20201110T213000Z
DTEND:20201110T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/15
DESCRIPTION:Title: Coherent Springer theory and categorical Deligne-Langlands\nby
Harrison Chen (Cornell University) as part of MIT Lie groups seminar\n\nLe
cture held in 2-142.\n\nAbstract\nKazhdan and Lusztig proved the Deligne-L
anglands conjecture\, a bijection between irreducible representations of u
nipotent principal block representations of a p-adic group with certain un
ipotent Langlands parameters in the Langlands dual group (plus the data of
certain representations). We lift this bijection to a statement on the
level of categories. Namely\, we define a stack of unipotent Langlands p
arameters and a coherent sheaf on it\, which we call the coherent Springer
sheaf\, which generates a subcategory of the derived category equivalent
to modules for the affine Hecke algebra (or specializing at q\, unipotent
principal block representations of a p-adic group). Our approach involve
s categorical traces\, Hochschild homology\, and Bezrukavnikov's Langlands
dual realizations of the affine Hecke category. This is a joint work wi
th David Ben-Zvi\, David Helm and David Nadler.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kostiantyn Tolmachov (University of Toronto)
DTSTART:20201104T213000Z
DTEND:20201104T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/16
DESCRIPTION:Title: Monodromic model for Khovanov-Rozansky homology\nby Kostiantyn
Tolmachov (University of Toronto) as part of MIT Lie groups seminar\n\nLec
ture held in 2-142.\n\nAbstract\nKhovanov-Rozansky homology is a knot inva
riant which\, by the result of Khovanov\, can be computed as the Hochschil
d cohomology functor applied Rouquier complexes of Soergel bimodules. I wi
ll describe a new geometric model for the Hochschild cohomology of Soergel
bimodules\, living in the monodromic Hecke category. I will also explain
how it allows to identify objects representing individual Hochsсhild coho
mology groups as images of explicit character sheaves.\n\nBased on the joi
nt work with Roman Bezrukavnikov.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yakov Kononov (Columbia University)
DTSTART:20201118T213000Z
DTEND:20201118T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/17
DESCRIPTION:Title: Elliptic stable envelopes and 3-dimensional mirror symmetry\nby
Yakov Kononov (Columbia University) as part of MIT Lie groups seminar\n\n
Lecture held in 2-142.\n\nAbstract\nThe action of quantum groups on the K-
theory of Nakajima varieties takes the simplest form in the stable bases\,
invented by D.Maulik and A.Okounkov\, and in their most advanced (ellipti
c) version by M.Aganagic and A.Okounkov. In collaboration with A.Smirnov w
e discovered and proved the factorization property of elliptic stable enve
lopes. As a consequence\, we proved the conjectures of E.Gorsky and A.Negu
t. Also it gives a new interesting description of the operators of quantum
difference equations\, shift operators and other quantities in enumerativ
e geometry. The talk is based on joint works with A.Smirnov.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Masao OI (University of Kyoto)
DTSTART:20201202T230000Z
DTEND:20201203T000000Z
DTSTAMP:20260404T094150Z
UID:MITLie/18
DESCRIPTION:Title: Twisted endoscopic character relation for Kaletha's regular supercu
spidal L-packets\nby Masao OI (University of Kyoto) as part of MIT Lie
groups seminar\n\nLecture held in 2-142.\n\nAbstract\nRecently Kaletha co
nstructed the local Langlands correspondence (i.e.\, L-packets and their L
-parameters) for a wide class of supercuspidal representations. In this
talk\, I would like to discuss my ongoing work on the twisted endoscopic c
haracter relation for Kaletha's supercuspidal L-packets.\n\nThe strategy i
s to imitate Kaletha's proof of the standard endoscopic character relation
in the setting of twisted endoscopy. Thus first I am going to review Ka
letha's construction of supercuspidal L-packets and his proof of the stand
ard endoscopic character relation. Then I will explain a few key points
in the twisting process with an emphasis on Waldspurger's philosophy "l'en
doscopic tordue n'est pas si tordue".\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Lusztig (MIT)
DTSTART:20201209T213000Z
DTEND:20201209T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/19
DESCRIPTION:Title: From families in Weyl groups to unipotent elements\nby George L
usztig (MIT) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n
\nAbstract\nIn geometric representation theory one tries to understand gro
up representations using geometry. But sometimes one can try to go in the
opposite direction. In this talk we will illustrate this by showing that a
number of features in geometry (such as Springer correspondence attached
to unipotent classes) can be recovered from pure algebra (such as the gene
ric degrees of representations of Weyl groups).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erez Lapid (Weizmann Institute)
DTSTART:20201216T213000Z
DTEND:20201216T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/20
DESCRIPTION:by Erez Lapid (Weizmann Institute) as part of MIT Lie groups s
eminar\n\nLecture held in 2-142.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Losev (Yale Universiy)
DTSTART:20210224T213000Z
DTEND:20210224T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/21
DESCRIPTION:Title: Unipotent Harish-Chandra bimodules\nby Ivan Losev (Yale Univers
iy) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstrac
t\nUnipotent representations of semisimple Lie groups is a very important
and somewhat conjectural class of unitary representations. Some of these r
epresentations for complex groups (equivalently\, Harish-Chandra bimodules
) were defined in the seminal paper of Barbasch and Vogan from 1985 based
on ideas of Arthur. From the beginning it was clear that the Barbasch-Voga
n construction doesn't cover all unipotent representations. The main const
ruction of this talk is a geometric construction of Harish-Chandra bimodul
es that should exhaust all unipotent bimodules. A nontrivial result is tha
t all unipotent bimodules in the sense of Barbasch and Vogan are also unip
otent in our sense. The proof of this claim is based on the so called symp
lectic duality that in our case upgrades a classical duality for nilpotent
orbits in the version of Barbasch and Vogan. Time permitting I will expla
in how this works. The talk is based on a joint work with Lucas Mason-Brow
n and Dmytro Matvieievskyi.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minh-Tam Trinh (MIT Mathematics)
DTSTART:20210303T213000Z
DTEND:20210303T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/22
DESCRIPTION:Title: From the Hecke Category to the Unipotent Locus\nby Minh-Tam Tri
nh (MIT Mathematics) as part of MIT Lie groups seminar\n\nLecture held in
2-142.\n\nAbstract\nWhen W is the Weyl group of a reductive group G\, we c
an categorify its Hecke algebra by means of equivariant sheaves on the dou
ble flag variety of G. We will define a functor from the resulting categor
y to a certain category of modules over a polynomial extension of C[W]. We
will prove that\, on objects called Rouquier complexes\, our functor yiel
ds the equivariant Borel-Moore homology of a generalized Steinberg variety
attached to a positive element in the braid group of W. Some reasons this
may be interesting: (1) In type A\, the triply-graded Khovanov-Rozansky h
omology of the link closure of the braid is a summand of the weight-graded
equivariant homology of this variety. This extends previously-known resul
ts for the top and bottom "a-degrees" of KR homology. (2) The "Serre duali
ty" of KR homology under insertion of full twists leads us to conjecture a
mysterious homeomorphism between pieces of different Steinbergs. (3) We f
ind evidence for a rational-DAHA action on the (modified) homology of the
Steinbergs of periodic braids. It seems related to conjectures of Broué-M
ichel and Oblomkov-Yun in rather different settings.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhilin Luo (University of Minnesota)
DTSTART:20210310T213000Z
DTEND:20210310T223000Z
DTSTAMP:20260404T094150Z
UID:MITLie/23
DESCRIPTION:Title: Harmonic analysis and gamma functions on symplectic group\nby Z
hilin Luo (University of Minnesota) as part of MIT Lie groups seminar\n\nL
ecture held in 2-142.\n\nAbstract\nWe develop a new type of harmonic analy
sis on an extended symplectic group $G=\\BG_m\\times \\Sp_2n$ over $p$-adi
c fields. It is associated with the Langlands $\\gamma$-functions attached
to irreducible admissible representations of $G(F)$ and the standard repr
esentation of the dual group. Our work can be viewed as an extension of th
e work of Godement-Jacquet (which is a generalization of Tate's thesis). W
e confirm a series of conjectures in the local theory of the Braverman-Kaz
hdan proposal in this setting. This is a joint work with D. Jiang and L. Z
hang.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ting Xue (University of Melbourne)
DTSTART:20210317T203000Z
DTEND:20210317T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/24
DESCRIPTION:Title: Graded Lie algebras\, character sheaves\, and representations of DA
HAs\nby Ting Xue (University of Melbourne) as part of MIT Lie groups s
eminar\n\nLecture held in 2-142.\n\nAbstract\nWe describe a strategy for c
lassifying character sheaves in the setting of graded Lie algebras. Via a
nearby cycle construction we show that irreducible representations of Heck
e algebras of complex reflection groups at roots of unity enter the descri
ption of character sheaves. We will explain connection to the work of Lusz
tig and Yun where (Fourier transforms of) character sheaves are parametriz
ed by irreducible representations of trigonometric double affine Hecke alg
ebras (DAHA). We will discuss some conjectures arising from this connectio
n\, which relate finite dimensional irreducible representations of trigono
metric DAHAs to irreducible representations of Hecke algebras. This is bas
ed on joint work with Kari Vilonen and partly with Misha Grinberg.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:German Stefanich (UC Berkeley)
DTSTART:20210324T203000Z
DTEND:20210324T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/25
DESCRIPTION:Title: Categorified sheaf theory and the spectral Betti Langlands TQFT
\nby German Stefanich (UC Berkeley) as part of MIT Lie groups seminar\n\nL
ecture held in 2-142.\n\nAbstract\nIt is expected that the Betti form of t
he geometric Langlands equivalence will ultimately fit into an equivalence
of four dimensional topological field theories. In this talk I will give
an overview of ongoing work in the theory of sheaves of higher categories
in derived algebraic geometry\, and explain how it can be used to define a
candidate four dimensional theory for the spectral side.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Mellit (University of Vienna)
DTSTART:20210407T203000Z
DTEND:20210407T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/26
DESCRIPTION:Title: Macdonald polynomials and counting parabolic bundles\nby Anton
Mellit (University of Vienna) as part of MIT Lie groups seminar\n\nLecture
held in 2-142.\n\nAbstract\nIt is well known that Hall-Littlewood polynom
ials naturally arise from the problem of counting partial flags preserved
by a nilpotent matrix over a finite field. I give an explicit interpretati
on of the modified Macdonald polynomials in a similar spirit\, via countin
g parabolic bundles with nilpotent endomorphism over a curve over finite f
ield. The result can also be interpreted as a formula for a certain trunca
ted weighted counting of points in the affine Springer fiber over a consta
nt nilpotent matrix. This leads to a confirmation of a conjecture of Hause
l\, Letellier and Rodriguez-Villegas about Poincare polynomials of charact
er varieties. On the other hand\, it naturally leads to interesting expans
ions of Macdonald polynomials and related generating functions that appear
in the shuffle conjecture and its generalizations.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vasily Krylov (MIT Mathematics)
DTSTART:20210414T203000Z
DTEND:20210414T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/27
DESCRIPTION:Title: Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric
Satake equivalence\nby Vasily Krylov (MIT Mathematics) as part of MIT
Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nThis talk is ba
sed on the paper (joint with M. Finkelberg and I. Mirković).\n\nLet G be
a reductive complex algebraic group. Recall that a geometric Satake isomor
phism is an equivalence between the category of G(O)-equivariant perverse
sheaves on the affine Grassmannian for G and the category of finite dimens
ional representations of the Langlands dual group \\hat{G}. It follows tha
t for any G(O)-equivariant perverse sheaf P there exists an action of the
dual Lie algebra \\hat{\\mathfrak{g}} on the global cohomology of P.\n\nWe
will explain one possible approach to constructing this action. To do so\
, we will describe a new geometric construction of the universal envelopin
g algebra of the positive nilpotent subalgebra of the Langlands dual Lie a
lgebra \\hat{\\mathfrak{g}} based on certain one-parametric deformation of
zastava spaces. We will introduce the so-called Drinfeld-Gaitsgory-Vinber
g interpolation Grassmannian that is a one-parametric deformation of the a
ffine Grassmannian Gr_G. We will discuss the case G=SL_2 as an example.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tsao-Hsien Chen (University of Minnesota)
DTSTART:20210421T203000Z
DTEND:20210421T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/28
DESCRIPTION:Title: Hitchin fibration and commuting schemes\nby Tsao-Hsien Chen (Un
iversity of Minnesota) as part of MIT Lie groups seminar\n\nLecture held i
n 2-142.\n\nAbstract\nThe commuting scheme has always been of great intere
st in invariant theory but it was only recent that it appears as a primord
ial object in the study of the Hitchin fibration for higher dimensional va
rieties. I will explain how the invariant theory for the commuting scheme\
, in particular the Chevalley restriction theorem for the commuting scheme
\, is used in the study of Hitchin fibration and the proof of the Chevalle
y restriction theorem in the case of symplectic Lie algebras. The talk is
based on joint work with Ngo Bao Chau.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucas Mason-Brown (University of Oxford)
DTSTART:20210428T203000Z
DTEND:20210428T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/29
DESCRIPTION:Title: What is a unipotent representation?\nby Lucas Mason-Brown (Univ
ersity of Oxford) as part of MIT Lie groups seminar\n\nLecture held in 2-1
42.\n\nAbstract\nThe concept of a unipotent representation has its origins
in the representation theory of finite Chevalley groups. Let G(Fq) be the
group of Fq-rational points of a connected reductive algebraic group G. I
n 1984\, Lusztig completed the classification of irreducible representatio
ns of G(Fq). He showed:\n\n1) All irreducible representations of G(Fq) can
be constructed from a finite set of building blocks -- called `unipotent
representations.'\n\n2) Unipotent representations can be classified by cer
tain geometric parameters related to nilpotent orbits for a complex group
associated to G(Fq).\n\nNow\, replace Fq with C\, the field of complex num
bers\, and replace G(Fq) with G(C). There is a striking analogy between th
e finite-dimensional representation theory of G(Fq) and the unitary repres
entation theory of G(C). This analogy suggests that all unitary representa
tions of G(C) can be constructed from a finite set of building blocks -- c
alled `unipotent representations' -- and that these building blocks are cl
assified by geometric parameters related to nilpotent orbits. In this talk
I will propose a definition of unipotent representations\, generalizing t
he Barbasch-Vogan notion of `special unipotent'. The definition I propose
is geometric and case-free. After giving some examples\, I will state a ge
ometric classification of unipotent representations\, generalizing the wel
l-known result of Barbasch-Vogan for special unipotents.\n\nThis talk is b
ased on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.\
n
LOCATION:https://stable.researchseminars.org/talk/MITLie/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shu-Yen Pan (National Tsinghua University (Taiwan))
DTSTART:20210505T203000Z
DTEND:20210505T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/30
DESCRIPTION:Title: Finite Howe correspondence and Lusztig classification\nby Shu-Y
en Pan (National Tsinghua University (Taiwan)) as part of MIT Lie groups s
eminar\n\nLecture held in 2-142.\n\nAbstract\nLet $(G\,G')$ be a reductive
dual pair inside a finite symplectic group. By restricting the Weil repre
sentation to the dual pair\, there exists a relation (called the finite Ho
we correspondence) between the irreducible representations of the two grou
ps $G\,G'$. In this talk\, we would like to discuss some progress on the u
nderstanding of the correspondence by using Lusztig's classification on th
e representations of finite classical groups.\n\nIn particular\, we will f
ocus on the following three subjects:\n1. the decomposition of the uniform
projection of the Weil character\n2. the commutativity between the Howe c
orrespondence and the Lusztig correspondence\n3. the description of the Ho
we correspondence on unipotent characters in terms of the symbols by Luszt
ig.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ben-Zvi (University of Texas - Austin)
DTSTART:20210512T203000Z
DTEND:20210512T213000Z
DTSTAMP:20260404T094150Z
UID:MITLie/31
DESCRIPTION:Title: Quantization and Duality for Spherical Varieties\nby David Ben-
Zvi (University of Texas - Austin) as part of MIT Lie groups seminar\n\nLe
cture held in 2-142.\n\nAbstract\nI will present joint work with Yiannis S
akellaridis and Akshay Venkatesh\, in which we apply a perspective from to
pological field theory to the relative Langlands program. To a spherical v
ariety one can assign two quantization problems\, automorphic and spectral
\, both resulting in structures borrowed from QFT. The automorphic quantiz
ation (or A-side) organizes objects such as periods\, Plancherel measure\,
theta series and relative trace formula\, while the spectral quantization
(or B-side) organizes L-functions and Langlands parameters. Our conjectur
es describe a duality operation on spherical varieties\, which exchanges a
utomorphic and spectral quantizations (and may be seen as Langlands dualit
y for boundary conditions in 4d TFT\, a refined form of symplectic duality
/ 3d mirror symmetry).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Gorelik (Weizmann Institute of Science)
DTSTART:20210519T173000Z
DTEND:20210519T183000Z
DTSTAMP:20260404T094150Z
UID:MITLie/32
DESCRIPTION:Title: Kac-Moody superalgebras and Duflo-Serganova functors\nby Maria
Gorelik (Weizmann Institute of Science) as part of MIT Lie groups seminar
\n\nLecture held in 2-142.\n\nAbstract\nThe central characters of the fini
te-dimensional Kac-Moody superalgebras can be described by their "cores"\;
this notion can be nicely interpreted in terms of the Duflo-Serganova fun
ctors. I will discuss an extension of these results to affine Lie superalg
ebras.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20210908T200000Z
DTEND:20210908T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/33
DESCRIPTION:Title: Constructing unipotent representations\nby David Vogan (MIT) as
part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nIn
the 1950s\, Mackey began a systematic analysis of unitary representations
of groups in terms of "induction" from normal subgroups. Ultimately this l
ed to a fairly good reduction of unitary representation theory to the case
of simple groups\, which lack interesting normal subgroups. At about the
same time\, Gelfand and Harish-Chandra understood that many representation
s of simple groups could be constructed using induction from parabolic sub
groups. After many refinements and extensions of this work\, there still r
emain a number of interesting representations of simple groups that are of
ten not obtained by parabolic induction.\n\nFor the case of real reductive
groups\, I will discuss a certain (finite) family of representations\, ca
lled unipotent\, whose existence was conjectured by Arthur in the 1980s. S
ome unipotent representations can in fact be obtained by parabolic inducti
on\; I will talk about when this ought to happen\, and about the (rather r
are) cases in which Arthur's unipotent representations are not induced. (A
lot of what I will say is meaningful and interesting over local or finite
fields\, but I know almost nothing about those cases.)\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wicher Malten (Oxford University)
DTSTART:20210915T200000Z
DTEND:20210915T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/34
DESCRIPTION:Title: From braids to transverse slices in reductive groups\nby Wicher
Malten (Oxford University) as part of MIT Lie groups seminar\n\nLecture h
eld in 2-142.\n\nAbstract\nWe explain how group analogues of Slodowy slice
s arise by interpreting certain Weyl group elements as braids. Such slices
originate from classical work by Steinberg on regular conjugacy classes\,
and different generalisations recently appeared in work by Sevostyanov on
quantum group analogues of W-algebras and in work by He-Lusztig on Delign
e-Lusztig varieties. Also building upon recent work of He-Nie\, our perspe
ctive furnishes a common generalisation and a simple geometric criterion f
or Weyl group elements to yield strictly transverse slices.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Lusztig (MIT)
DTSTART:20210922T200000Z
DTEND:20210922T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/35
DESCRIPTION:Title: Total positivity in symmetric spaces\nby George Lusztig (MIT) a
s part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nTh
e theory of total positive matrices in GL_n(R) was initiated by Schoenberg
(1930) and Gantmacher-Krein (1935) and extended to reductive groups in my
1994 paper. It turns out that much of the theory makes sense also for sym
metric spaces although some new features arise.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Losev (Yale)
DTSTART:20210929T200000Z
DTEND:20210929T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/36
DESCRIPTION:Title: Harish-Chandra modules over quantizations of nilpotent orbits.\
nby Ivan Losev (Yale) as part of MIT Lie groups seminar\n\nLecture held in
2-142.\n\nAbstract\nLet O be a nilpotent orbit in a semisimple Lie algebr
a over the complex numbers. Then it makes sense to talk about filtered qua
ntizations of O\, these are certain associative algebras that necessarily
come with a preferred homomorphism from the universal enveloping algebra.
Assume that the codimension of the boundary of O is at least 4\, this is t
he case for all birationally rigid orbits (but six in the exceptional type
)\, for example. In my talk I will explain a geometric classification of f
aithful irreducible Harish-Chandra modules over quantizations of O\, conce
ntrating on the case of canonical quantizations -- this gives rise to mod
ules that could be called unipotent. The talk is based on a joint paper wi
th Shilin Yu (in preparation).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xuhua He (Chinese U. Hong Kong)
DTSTART:20211006T140000Z
DTEND:20211006T150000Z
DTSTAMP:20260404T094150Z
UID:MITLie/37
DESCRIPTION:Title: Frobenius-twisted conjugacy classes of loop groups and Demazure pro
duct of Iwhaori-Weyl groups\nby Xuhua He (Chinese U. Hong Kong) as par
t of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nThe aff
ine Deligne-Lusztig varieties\, roughly speaking\,\ndescribe the intersect
ion of Iwahori-double cosets and Frobenius-twisted\nconjugacy classes in a
loop group. For each fixed Iwahori-double coset\n$I w I$\, there exists a
unique Frobenius-twisted conjugacy class whose\nintersection with $I w I$
is open dense in $I w I$. Such\nFrobenius-twisted conjugacy class $[b_w]$
is called the generic\nFrobenius-twisted conjugacy class with respect to
the element $w$.\nUnderstanding $[b_w]$ leads to some important consequenc
es in the study\nof affine Deligne-Lusztig varieties. In this talk\, I wil
l give an\nexplicit description of $[b_w]$ in terms of Demazure product of
the\nIwahori-Weyl groups. It is worth pointing out that a priori\, $[b_w]
$ is\nrelated to the conjugation action on $I w I$\, and it is interesting
that\n$[b_w]$ can be described using Demazure product instead of conjugat
ion\naction. This is based on my preprint arXiv:2107.14461.\n\nIf time all
ows\, I will also discuss an interesting application. Lusztig\nand Vogan r
ecently introduced a map from the set of translations to the\nset of domin
ant translations in the Iwahori-Weyl group. As an\napplication of the conn
ection between $[b_w]$ and Demazure product\, we\nwill give an explicit fo
rmula for the map of Lusztig and Vogan.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yaping Yang (U. Melbourne)
DTSTART:20211020T200000Z
DTEND:20211020T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/38
DESCRIPTION:Title: Frobenii on Morava E-theoretical quantum groups\nby Yaping Yang
(U. Melbourne) as part of MIT Lie groups seminar\n\nLecture held in 2-142
.\n\nAbstract\nIn this talk\, I will explain a connection between stable h
omotopy theory and representation theory. I will focus on one application
of this idea to a problem arising from the modular representation theory.
More explicitly\, we study a family of new quantum groups labelled by a pr
ime number and a positive integer constructed using the Morava E-theories.
Those quantum groups are related to Lusztig's 2015 reformulation of his c
onjecture from 1979 on character formulas for algebraic groups over a fiel
d of positive characteristic. This talk is based on my joint work with Guf
ang Zhao.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Berest (Cornell)
DTSTART:20211027T200000Z
DTEND:20211027T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/39
DESCRIPTION:Title: Topological realization of rings of quasi-invariants of finite refl
ection groups\nby Yuri Berest (Cornell) as part of MIT Lie groups semi
nar\n\n\nAbstract\nQuasi-invariants are natural geometric generalizations
of classical invariant polynomials of finite reflection groups. They first
appeared in mathematical physics in the early 1990s\, and since then have
found applications in a number of other areas (most notably\, representat
ion theory\, algebraic geometry and combinatorics).\n\nIn this talk\, I wi
ll explain how the algebras of quasi-invariants can be realized topologica
lly: as (equivariant) cohomology rings of certain spaces naturally attache
d to compact connected Lie groups. Our main result can be viewed as a gene
ralization of a well-known theorem of A. Borel that realizes the algebra o
f invariant polynomials of a Weyl group W as the cohomology ring of the cl
assifying space BG of the corresponding Lie group G. Replacing equivariant
cohomology with equivariant K-theory gives a multiplicative (exponential)
analogues of quasi-invariants of Weyl groups. But perhaps more interestin
g is the fact that one can also realize topologically the quasi-invariants
of some non-Coxeter groups: our `spaces of quasi-invariants' can be const
ructed in a purely homotopy-theoretic way\, and this construction extends
naturally to (p-adic) pseudoreflection groups. In this last case\, the com
pact Lie groups are replaced by p-compact groups (a.k.a. homotopy Lie grou
ps). The talk is based on joint work with A. C. Ramadoss.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milen Yakimov (Northeastern University)
DTSTART:20211110T210000Z
DTEND:20211110T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/40
DESCRIPTION:Title: Quantum symmetric pairs via star products\nby Milen Yakimov (No
rtheastern University) as part of MIT Lie groups seminar\n\nLecture held i
n 2-142.\n\nAbstract\nThe systematic study of quantum symmetric pairs (QSP
s) was initiated by Gail Letzter in 1999. The area has been greatly develo
ped in recent years. We will present a new approach to the theory of quant
um symmetric pairs for symmetrizable Kac-Moody algebras based on star prod
ucts on noncommutative graded algebras. It will be used to give solutions
to two main problems in the area: (1) determine the defining relations of
QSPs and (2) find a Drinfeld type formula for universal $K$-matrices as su
ms of tensor products over dual bases. This is a joint work with Stefan Ko
lb.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART:20211013T200000Z
DTEND:20211013T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/41
DESCRIPTION:Title: Derived Chevalley isomorphisms\nby Tony Feng (MIT) as part of M
IT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nFor a reducti
ve group G\, the classical Chevalley isomorphism identifies conjugation-in
variant functions on G with Weyl-invariant functions on its maximal torus.
Berest-Ramadoss-Yeung have conjectured a derived upgrade of this statemen
t\, which predicts that the conjugation-invariant functions on the derived
commuting variety of G identify with the Weyl-invariant functions on the
derived commuting variety of its maximal torus. In joint work with Dennis
Gaitsgory we deduce this conjecture for G = GL_n from investigations into
derived aspects of the local Langlands correspondence. I’ll explain this
story\, assuming no background in derived algebraic geometry.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Crooks (Northeastern University)
DTSTART:20211103T200000Z
DTEND:20211103T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/42
DESCRIPTION:Title: Universal symplectic quotients via Lie theory\nby Peter Crooks
(Northeastern University) as part of MIT Lie groups seminar\n\nLecture hel
d in 2-142.\n\nAbstract\nIn its most basic form\, symplectic geometry is a
mathematically rigorous framework for classical mechanics. Noether's pers
pective on conserved quantities thereby gives rise to quotient constructio
ns in symplectic geometry. The most classical such construction is Marsden
-Weinstein-Meyer reduction\, while more modern variants include Ginzburg-K
azhdan reduction\, Kostant-Whittaker reduction\, Mikami-Weinstein reductio
n\, symplectic cutting\, and symplectic implosion.\n\nI will provide a sim
ultaneous generalization of the quotient constructions mentioned above. Th
is generalization will be shown to have versions in the smooth\, holomorph
ic\, complex algebraic\, and derived symplectic contexts. As a corollary\,
I will derive a concrete and Lie-theoretic construction of "universal" sy
mplectic quotients.\n\nThis represents joint work with Maxence Mayrand.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Braverman (University of Toronto)
DTSTART:20211117T210000Z
DTEND:20211117T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/43
DESCRIPTION:Title: Examples of Hecke eigen-functions for moduli spaces of bundles over
local non-archimedean field and an analog of Eisenstein series\nby Al
exander Braverman (University of Toronto) as part of MIT Lie groups semina
r\n\nLecture held in 2-142.\n\nAbstract\nLet X be a smooth projective curv
e over a finite field $k$\, and let $G$ be a reductive group. The unramifi
ed part of the theory of automorphic forms for the group G and the field $
k(X)$ studies functions on the $k$-points on the moduli space of $G$-bundl
es on $X$ and the eigen-functions of the Hecke operators (to be reviewed i
n the talk!) acting there. The spectrum of the Hecke operators has continu
ous and discrete parts and it is described by the global Langlands conject
ures (which in the case of functional fields are essentially proved by V.L
afforgue).\n\nAfter recalling the above notions and constructions I will d
iscuss what happens when $k$ is replaced by a local field. The correspondi
ng Hecke operators were essentially defined by myself and Kazhdan about 10
years ago\, but the systematic study of eigen-functions has begun only re
cently. It was initiated several years ago by Langlands when $k$ is archim
edean and then Etingof\, Frenkel and Kazhdan formulated a very precise con
jecture describing the spectrum in terms of the dual group. Contrary to th
e classical case only discrete spectrum is expected to exist. I will discu
ss what is is known in the case when $k$ is a local non-archimedean field
$K$. In particular\, I will talk about some version of the Eisenstein seri
es operator which allows to construct a Hecke eigen-function over $K$ star
ting from a cuspidal Hecke eigen-function over finite field (joint work i
n progress with D.Kazhdan and A.Polishchuk).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan)
DTSTART:20211201T210000Z
DTEND:20211201T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/44
DESCRIPTION:Title: Seminar Cancelled\nby Tasho Kaletha (University of Michigan) as
part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\nSem
inar Cancelled\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Ciubotaru (Oxford University)
DTSTART:20211208T210000Z
DTEND:20211208T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/45
DESCRIPTION:Title: A nonabelian Fourier transform for tempered unipotent representatio
ns of p-adic groups\nby Dan Ciubotaru (Oxford University) as part of M
IT Lie groups seminar\n\nLecture held in The Simons Building in Room: 2-14
2.\n\nAbstract\nIn the representation theory of finite reductive groups\,
an essential role is played by Lusztig's nonabelian Fourier transform\, an
involution on the space of unipotent characters the group. This involutio
n is the change of bases matrix between the basis of irreducible character
s and the basis of `almost characters'\, certain class functions attached
to character sheaves. For reductive p-adic groups\, the unipotent local La
nglands correspondence gives a natural parametrization of irreducible smoo
th representations with unipotent cuspidal support. However\, many questio
ns about the characters of these representations are still open. Motivated
by the study of the characters on compact elements\, we introduce in join
t work with A.-M. Aubert and B. Romano (arXiv:2106.13969) an involution on
the spaces of elliptic and compact tempered unipotent representations of
pure inner twists of a split simple p-adic group. This generalizes a const
ruction by Moeglin and Waldspurger (2003\, 2016) for elliptic tempered rep
resentations of split orthogonal groups\, and potentially gives another in
terpretation of a Fourier transform for p-adic groups introduced by Luszti
g (2014). We conjecture (and give supporting evidence) that the restrictio
n to reductive quotients of maximal compact open subgroups intertwines thi
s involution with a disconnected version of Lusztig's nonabelian Fourier t
ransform for finite reductive groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kari Vilonen (Melbourne)
DTSTART:20220209T210000Z
DTEND:20220209T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/46
DESCRIPTION:Title: Mixed Hodge modules and representation theory of real groups\nb
y Kari Vilonen (Melbourne) as part of MIT Lie groups seminar\n\nLecture he
ld in 2-142.\n\nAbstract\n\\noindent I will explain how mixed Hodge module
s can be utilized to understand representation theory of real groups. In p
articular\, we obtain a refinement of the Lusztig-Vogan polynomials in thi
s setting. Adams\, van Leeuwen\, Trapa\, and Vogan (ALTV) have given an al
gorithm to determine the unitary dual of a real reductive group. As a coro
llary of our results we obtain a proof of a key result of (ALTV) on signat
ure polynomials. \\\\\n\\vspace{2ex}\n\\noindent This is joint work with D
ougal Davis.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emile Okada (University of Oxford)
DTSTART:20220216T210000Z
DTEND:20220216T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/47
DESCRIPTION:Title: The wavefront set and Arthur packets of p-adic groups\nby Emile
Okada (University of Oxford) as part of MIT Lie groups seminar\n\nLecture
held in 2-142.\n\nAbstract\nThe wavefront set is a powerful harmonic anal
ytic invariant attached to representations of p-adic groups that is expect
ed to play an important role in the construction of Arthur packets. In thi
s talk I will present new results relating it to the local Langlands corre
spondence for representations in the principal block. In the process I wil
l introduce a natural refinement of the (geometric) wavefront set with man
y nicer properties and use it to construct some unipotent Arthur packets o
f arbitrary split groups. The results are based on joint work with Dan Ciu
botaru and Lucas Mason-Brown.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuchen Fu (Harvard University)
DTSTART:20220223T210000Z
DTEND:20220223T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/48
DESCRIPTION:Title: Kazhdan-Lusztig Equivalence at the Iwahori Level\nby Yuchen Fu
(Harvard University) as part of MIT Lie groups seminar\n\nLecture held in
2-142.\n\nAbstract\nWe construct an equivalence between Iwahori-integrable
representations of affine Lie algebras and representations of the "mixed"
quantum group\, thus confirming a conjecture by Gaitsgory. Our proof util
izes factorization methods: we show that both sides are equivalent to alge
braic/topological factorization modules over a certain factorization algeb
ra\, which can then be compared via Riemann-Hilbert. On the quantum group
side this is achieved via general machinery of homotopical algebra\, where
as the affine side requires inputs from the theory of (renormalized) ind-c
oherent sheaves as well as compatibility with global Langlands over P1.\n\
nThis is joint work with Lin Chen.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan)
DTSTART:20220302T210000Z
DTEND:20220302T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/49
DESCRIPTION:Title: Characterization and construction of the local Langlands correspond
ence for supercuspidal parameters\nby Tasho Kaletha (University of Mic
higan) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbst
ract\nWe will formulate a list of properties that uniquely characterize th
e local Langlands correspondence for discrete Langlands parameters with tr
ivial monodromy. Suitably interpreted\, this characterization holds for an
y local field\, but requires an assumption on p in the non-archimedean cas
e. We will then discuss an explicit construction of this correspondence\,
as a realization of functorial transfer from double covers of elliptic max
imal tori.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cedric Bonnafe (CNRS)
DTSTART:20220309T210000Z
DTEND:20220309T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/50
DESCRIPTION:Title: Calogero-Moser spaces vs unipotent representations\nby Cedric B
onnafe (CNRS) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\
n\nAbstract\nTitle to be shared\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Negut (MIT)
DTSTART:20220316T200000Z
DTEND:20220316T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/51
DESCRIPTION:Title: On the trace of the affine Hecke category\nby Andrei Negut (MIT
) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\
nWe propose a connection between the horizontal trace of the affine Hecke
category and the elliptic Hall algebra\, mirroring known constructions for
the finite Hecke category. Explicitly\, we construct a family of generato
rs of the affine Hecke category\, compute certain categorified commutators
between them\, and show that their K-theoretic shadows match certain comm
utators in the elliptic Hall algebra. Joint work with Eugene Gorsky.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Gannon (University of Texas)
DTSTART:20220330T200000Z
DTEND:20220330T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/52
DESCRIPTION:Title: Categorical Representation Theory and the Coarse Quotient\nby T
om Gannon (University of Texas) as part of MIT Lie groups seminar\n\nLectu
re held in 2-142.\n\nAbstract\nThe main theorem of this talk will be that
one can understand a "dense open" subset of DG categories with an action o
f a split reductive group G over a field of characteristic zero entirely i
n terms of its root datum. We will start by introducing the notion of a ca
tegorical representation of G and discuss some motivation. Then\, we will
discuss some of the main technical tools involved in making the statement
of the main theorem precise\, including discussion of the "coarse quotient
" of the dual maximal Cartan by the affine Weyl group. We will also discus
s how sheaves on this coarse quotient can be identified with bi-Whittaker
sheaves on G\, obtaining symmetric monoidal upgrade of a result of Ginzbur
g and Lonergan\, and then give an outline of the proof of the main theorem
. Time permitting\, we will discuss some applications of these categorical
representation theoretic ideas which prove a modified version of a conjec
ture of Ben-Zvi and Gunningham on the essential image of parabolic restric
tion.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Caltech)
DTSTART:20220406T200000Z
DTEND:20220406T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/53
DESCRIPTION:Title: Perverse mod p sheaves on affine flag varieties\nby Robert Cass
(Caltech) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\n
Abstract\nPerverse sheaves have important applications in representation t
heory and number theory. In this talk we will consider the case of mod p
étale sheaves on affine flag varieties over a field of characteristic p.
Despite the pathological behavior of such sheaves\, they encode the struct
ure of mod p Hecke algebras. We will primarily focus on a version of the g
eometric Satake equivalence for the affine Grassmannian. Time permitting\,
we may also discuss central sheaves on the Iwahori affine flag variety. P
art of this is joint work with Cédric Pépin.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Dillery (University of Michigan)
DTSTART:20220504T200000Z
DTEND:20220504T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/54
DESCRIPTION:Title: Title to be announced\nby Peter Dillery (University of Michigan
) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nAbstract\
nTitle to be shared\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pramod Achar (LSU)
DTSTART:20220511T200000Z
DTEND:20220511T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/55
DESCRIPTION:Title: Co-t-structures on coherent sheaves and the Humphreys conjecture\nby Pramod Achar (LSU) as part of MIT Lie groups seminar\n\nLecture held
in 2-142.\n\nAbstract\nLet G be a connected reductive group over an algeb
raically closed field\, and let C be a nilpotent orbit for G. If L is an
irreducible G-equivariant vector bundle on C\, then one can define a "cohe
rent intersection cohomology complex" IC(C\,L). These objects play an impo
rtant role in various results related to the local geometric Langlands pro
gram. \n\nWhen G has positive characteristic\, instead of an irreducible b
undle L\, one might consider a tilting bundle T on C. I will explain a ne
w construction that associates to the pair (C\,T) a complex of coherent sh
eaves S(C\,T) with remarkable Ext-vanishing properties. This construction
leads to a proof of a conjecture of Humphreys on (relative) support varie
ties for tilting modules\, and hints at a kind of "recursive" structure in
the tensor category of tilting G-modules. This work is joint with W. Har
desty (and also partly with S. Riche).\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Ionov (MIT)
DTSTART:20220413T200000Z
DTEND:20220413T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/56
DESCRIPTION:Title: Tilting sheaves for real groups and Koszul duality\nby Andrei I
onov (MIT) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\n
Abstract\nFor a real form of an algebraic group acting on the flag variety
we define a t-structure on the category of equivariant-monodromic sheaves
and develop the theory of tilting sheaves. In case of a quasi-split real
form we construct an analog of a Soergel functor\, which full-faithfully e
mbeds the subcategory of tilting objects to the category of coherent sheav
es on a block variety. We apply the results to give a new\, purely geometr
ic\, proof of the Soergel's conjecture for quasi-split groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Sommers (UMass)
DTSTART:20220420T200000Z
DTEND:20220420T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/57
DESCRIPTION:Title: Hessenberg varieties and the geometric modular law\nby Eric Som
mers (UMass) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n
\nAbstract\nHessenberg varieties are fibers of certain proper maps to a si
mple Lie algebra. These maps are generalizations of the Springer and Groth
endieck-Springer resolutions. In this talk\, we describe some new properti
es of nilpotent Hessenberg varieties. In particular\, we show that their c
ohomology satisfies a modular law as we vary the maps. This law generalize
s one of De Concini\, Lusztig\, and Procesi and coincides with a combinato
rial law of Guay-Paquet and Abreu-Nigro in type A. We also study the push-
forward of the constant sheaf of these maps and show that only intersectio
n cohomology sheaves with local systems coming from the Springer correspon
dence appear in the decomposition\, resolving a conjecture of Brosnan. Thi
s is joint work with Martha Precup.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Ciubotaru (University of Oxford)
DTSTART:20220921T200000Z
DTEND:20220921T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/58
DESCRIPTION:Title: Wavefront sets and unipotent representations of p-adic groups\n
by Dan Ciubotaru (University of Oxford) as part of MIT Lie groups seminar\
n\nLecture held in 2-142.\n\nAbstract\nAn important invariant for admissib
le representations of reductive p-adic groups is the wavefront set\, the c
ollection of the maximal nilpotent orbits in the support of the orbital in
tegrals that occur in the Harish-Chandra-Howe local character expansion. W
e compute the geometric and Okada's canonical unramified wavefront sets fo
r representations in Lusztig's category of unipotent reduction for a split
group in terms of the Kazhdan-Lusztig parameters. We use this calculation
to give a new characterisation of the anti-tempered unipotent Arthur pack
ets. Another interesting consequence is that the geometric wavefront set o
f a unipotent supercuspidal representation uniquely determines the nilpote
nt part of the Langlands parameter\; this is an extension to p-adic groups
of Lusztig's result for unipotent representations of finite groups of Lie
type. The talk is based on joint work with Lucas Mason-Brown and Emile Ok
ada.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raphael Beuzart-Plessis (Marseille)
DTSTART:20220928T200000Z
DTEND:20220928T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/59
DESCRIPTION:Title: Title to be announced\nby Raphael Beuzart-Plessis (Marseille) a
s part of MIT Lie groups seminar\n\nLecture held in 2-142.\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/MITLie/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Oblomkov (U. Mass)
DTSTART:20221005T200000Z
DTEND:20221005T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/60
DESCRIPTION:Title: Affine Springer fibers and sheaves on Hilbert scheme of points on t
he plane.\nby Alexei Oblomkov (U. Mass) as part of MIT Lie groups semi
nar\n\nLecture held in 2-142 in the Simons building.\n\nAbstract\nMy talk
is based on the joint work with E. Gorsky and O. Kivinen.\nI will explain
a construction that associates a coherent sheaf on the\nHilbert scheme of
points on the plane to plane curve singularity. The\nglobal sections of th
e sheaf are equal to cohomology of the\ncorresponding Affine (type A) Spr
inger fiber. The construction\ncategorifies HOMFLYPT homology/cohomogy of
compactified Jacobian\nconjecture if combined with Soergel bimodule/ Shea
ves of Hilbert\nscheme theorem of Oblomkov-Rozansky. I will also discuss\n
generalizations outside of type A.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dennis Gaitsgory (Bonn)
DTSTART:20221012T200000Z
DTEND:20221012T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/61
DESCRIPTION:Title: From geometric Langlands to classical via the trace of Frobenius\nby Dennis Gaitsgory (Bonn) as part of MIT Lie groups seminar\n\nLecture
held in 2-142.\n\nAbstract\nI'll start by summarizing the main results of
the series [AGKRRV]\, where it is shown that the trace of Frobenius on th
e category of automorphic sheaves with nilpotent singular support identifi
es with the space of unramified automorphic functions. We'll then discuss
conjectural counterparts of this statement in the local and global ramifie
d settings.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiao Wang (University of Chicago)
DTSTART:20221019T200000Z
DTEND:20221019T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/62
DESCRIPTION:Title: Title to be announced\nby Xiao Wang (University of Chicago) as
part of MIT Lie groups seminar\n\nLecture held in 2-142.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jialiang Zou (University of Michigan)
DTSTART:20221026T200000Z
DTEND:20221026T210000Z
DTSTAMP:20260404T094150Z
UID:MITLie/63
DESCRIPTION:Title: On some Hecke algebra modules arising from theta correspondence and
it’s deformation\nby Jialiang Zou (University of Michigan) as part
of MIT Lie groups seminar\n\nLecture held in 2-142 in the Simons building.
\n\nAbstract\nThis talk is based on the joint work with Jiajun Ma and Cong
ling Qiu on theta correspondence of type I dual pairs over a finite field
$F_q$. We study the Hecke algebra modules arising from theta corresponden
ce between certain Harish-Chandra series for these dual pairs. We first sh
ow that the normalization of the corresponding Hecke algebra is related t
o the first occurrence index\, which leads to a proof of the conservation
relation. We then study the deformation of this Hecke algebra module at q
=1 and generalize the results of Aubert-Michel-Rouquier and Pan on theta c
orrespondence between unipotent representations along this way.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joel Kamnitzer (University of Toronto)
DTSTART:20221109T210000Z
DTEND:20221109T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/64
DESCRIPTION:Title: Moduli space of flower curves\nby Joel Kamnitzer (University of
Toronto) as part of MIT Lie groups seminar\n\nLecture held in 2-142.\n\nA
bstract\nThe Deligne-Mumford moduli space of genus 0 curves plays many rol
es in representation theory. For example\, the fundamental group of its re
al locus is the cactus group which acts on tensor products of crystals. I
will discuss a variant on this space which parametrizes "flower curves".
The fundamental group of the real locus of this space is the virtual cactu
s group. This moduli space of flower curves is also the parameter space fo
r inhomogeneous Gaudin algebras.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Mautner (UC Riverside)
DTSTART:20221116T210000Z
DTEND:20221116T220000Z
DTSTAMP:20260404T094150Z
UID:MITLie/65
DESCRIPTION:Title: Perverse sheaves on symmetric products of the plane\nby Carl Ma
utner (UC Riverside) as part of MIT Lie groups seminar\n\nLecture held in
2-142.\n\nAbstract\nIn joint work with Tom Braden we give a purely algebra
ic description of the category of perverse sheaves (with coefficients in a
ny field) on $S^n(C^2)$\, the n-fold symmetric product of the plane. In p
articular\, using the geometry of the Hilbert scheme of points\, we relate
this category to the symmetric group and its representation ring. Our wo
rk is motivated by analogous structure appearing in the Springer resolutio
n and Hilbert-Chow morphism.\n
LOCATION:https://stable.researchseminars.org/talk/MITLie/65/
END:VEVENT
END:VCALENDAR