BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220106T153000Z
DTEND:20220106T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/1
DESCRIPTION:Title: What groups? What representations? What software?\nby David Vogan
(MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nThe atlas sof
tware is about a connected reductive algebraic group $G$ over the field ${
\\mathbb R}$ of real numbers\, with group of real points $G({\\mathbb R})$
. It is possible to tell the software to consider ANY such $G({\\mathbb R}
)$\, of rank up to 32\, subject to memory size limitations in the computer
. (The distinction between $G$ and $G({\\mathbb R})$ is of fundamental the
oretical importance\, but inside the software we will often ignore it.) In
practice\, the software recognizes many more or less standard names for i
mportant examples of real $G({\\mathbb R})$\, and one can just learn to us
e those.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220113T153000Z
DTEND:20220113T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/2
DESCRIPTION:Title: Root data: how atlas understands a reductive group\nby Jeffrey Ad
ams (University of Maryland) as part of Real reductive groups/atlas\n\n\nA
bstract\nA root datum is a discrete mathematics object which (by the work
of Cartan-Killing\, Chevalley\, and Grothendieck) can be used to specify a
reductive algebraic group $G$ over any algebraically closed field $k$. Th
is is the starting point for how the ${\\tt atlas}$ software can work with
a real reductive group.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220120T153000Z
DTEND:20220120T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/3
DESCRIPTION:Title: Parameters: how atlas understands a representation\nby David Voga
n (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nThe represen
tation theory of a real reductive group $G$ is parallel to the theory of V
erma modules. There is a (concrete\, not too complicated) set of ``Langlan
ds parameters" for $G$. For each Langlands parameter there is a standard r
epresentation $I(p)$\, analogous to a Verma module: (relatively) easy to c
onstruct and understand. Each standard representation has a unique irreduc
ible (Langlands) quotient representation $J(p)$\, which can be a small and
subtle part of $I(p)$. Listing irreducible representations is therefore f
airly easy\; understanding them is harder.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220127T153000Z
DTEND:20220127T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/4
DESCRIPTION:Title: Branching to $K$: writing an infinite-dimensional representation of $
G$ as a sum of finite-dimensionals of $K$\nby Jeffrey Adams (Universit
y of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nThis
is a topic in which we can compute almost anything\, but where our underst
anding of the answers is much more limited. That makes it an excellent top
ic for research!\n
LOCATION:https://stable.researchseminars.org/talk/atlas/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220203T153000Z
DTEND:20220203T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/5
DESCRIPTION:Title: Understanding $K$: how atlas understands Cartan's theory of maximal c
ompact subgroups\nby David Vogan (MIT) as part of Real reductive group
s/atlas\n\n\nAbstract\nTo study an algebraic variety $X$ over a finite fie
ld ${\\mathbb F}_q$\, we use the Frobenius map $F$\, an algebraic map from
$X$ to $X$ whose set of fixed points is precisely $X({\\mathbb F}_q)$. Th
e power of this tool stems from the fact that $F$ (unlike the Galois group
action) is algebraic.\n\nTo study a reductive algebraic group $G$ over ${
\\mathbb R}$\, Cartan introduced the Cartan involution $\\theta$\, an alge
braic automorphism of $G$ having order 2. The group of fixed points of $\\
theta$ is an algebraic subgroup $K$ of $G$. It is not true (as in the fini
te field case) that $K$ is equal to $G({\\mathbb R})$\, but Cartan showed
how to get a great deal of information about $G({\\mathbb R})$ from $K$. T
his idea is at the heart of almost all that ${\\tt atlas}$ does.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220210T153000Z
DTEND:20220210T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/6
DESCRIPTION:Title: The Atlas Way (More on KGB)\nby Jeffrey Adams (University of Mary
land) as part of Real reductive groups/atlas\n\n\nAbstract\nExplain the no
tion of "strong real form\," which is central to how atlas writes down Lan
glands parameters.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220217T153000Z
DTEND:20220217T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/7
DESCRIPTION:Title: Character formulas and Kazhdan-Lusztig polynomials (or why we needed
this software in the first place)\nby David Vogan (MIT) as part of Rea
l reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220224T153000Z
DTEND:20220224T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/8
DESCRIPTION:Title: Signature character formulas and unitary representations (or why we n
eeded this software in the first place)\nby Jeffrey Adams (University
of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nWe firs
t discuss how it might be possible to write a finite formula for the signa
ture of a hermitian form on an infinite-dimensional space. Then we explain
how atlas computes that signature.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220303T153000Z
DTEND:20220303T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/9
DESCRIPTION:Title: Signature character formulas and unitary representations 2\nby Je
ffrey Adams (University of Maryland) as part of Real reductive groups/atla
s\n\n\nAbstract\nMore details about how atlas computes signatures: realize
a representation as a Langlands quotient J(delta\\otimes nu)\, then defor
m nu to zero. At nu=0\, the form is unitary. As nu changes\, the signature
of the form changes only at the (finitely many) values of t*nu (t in [0\,
1]) for which I(delta\\otimes t*nu) is reducible.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220317T143000Z
DTEND:20220317T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/10
DESCRIPTION:Title: Weyl group representations and atlas\nby Jeffrey Adams (Universi
ty of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nMuch
of the structure and representation theory of a reductive algebraic group
G is described using the Weyl group W. This talk will introduce what atla
s knows about the structure and representations of W.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220331T143000Z
DTEND:20220331T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/11
DESCRIPTION:Title: Gelfand-Kirillov dimension and atlas\nby David Vogan (MIT) as pa
rt of Real reductive groups/atlas\n\n\nAbstract\nLinear algebra wants to b
e about diagonal matrices. Nilpotent matrices control exactly how this des
ire must be frustrated\, and what is actually true instead. In a precisely
parallel way\, the theory of reductive groups wants to be about maximal t
ori. Nilpotent classes (for example in the Lie algebra) control how this d
esire must be frustrated\, and what is actually true instead. The talk wil
l concern what atlas knows about nilpotent Lie algebra elements\, with per
haps some small hints about what this information does for representation
theory.\n\nSeminar was actually all about Gelfand-Kirillov dimension: the
reason is that this is a really elementary and interesting invariant that
you can only hope to compute using nilpotent orbits.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220310T153000Z
DTEND:20220310T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/12
DESCRIPTION:Title: How atlas does what it says its doing\nby David Vogan (MIT) as p
art of Real reductive groups/atlas\n\n\nAbstract\nOften the descriptions w
e've sketched of the mathematical objects that atlas studies are very clos
e to their internal representations in the software. I'll talk about two s
ituations where that's not exactly the case: Fokko du Cloux's compact repr
esentation of Weyl group elements\, and the replacement of parameters by t
he "facets" in which they lie.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220324T143000Z
DTEND:20220324T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/13
DESCRIPTION:Title: Weyl group representations and atlas II\nby Jeffrey Adams (Unive
rsity of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nC
ontinuation of last week's seminar. Particular topics to include branching
of representations from a Weyl group to a subgroup\; motivation of the de
finition of coherent families\; relationship between Lusztig's "families"
in W^ and cell representations of W.\n\nIf time permits\, David Vogan will
do the second half of this seminar\, on the topic of Gelfand-Kirillov dim
ension (meant as an introduction to the nilpotent orbit talk next week).\n
LOCATION:https://stable.researchseminars.org/talk/atlas/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220407T143000Z
DTEND:20220407T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/14
DESCRIPTION:Title: Nilpotent orbits and atlas\nby David Vogan (MIT) as part of Real
reductive groups/atlas\n\n\nAbstract\nLinear algebra wants to be about di
agonal matrices. Nilpotent matrices control exactly how this desire must b
e frustrated\, and what is actually true instead. In a precisely parallel
way\, the theory of reductive groups wants to be about maximal tori. Nilpo
tent classes (for example in the Lie algebra) control how this desire must
be frustrated\, and what is actually true instead. The talk will concern
what atlas knows about nilpotent Lie algebra elements. The goal is to expl
ain how this helps with the computation of Gelfand-Kirillov dimension.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220414T143000Z
DTEND:20220414T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/15
DESCRIPTION:Title: Cell representations continued\nby Jeffrey Adams (University of
Maryland) as part of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220421T143000Z
DTEND:20220421T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/16
DESCRIPTION:Title: Lusztig's parametrization of families\nby Jeffrey Adams (Univers
ity of Maryland) as part of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220428T143000Z
DTEND:20220428T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/17
DESCRIPTION:Title: Real parabolic subgroups and induction in atlas\nby David Vogan
(MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nThe oldest con
struction of irreducible unitary representactions of a real reductive grou
p G(R) is unitary induction from a real parabolic subgroup P(R). A bit mor
e precisely\, P(R) has a well-defined normal subgroup U(R)\, the unipotent
radical\; and the quotient L(R) = P(R)/U(R) is again a real reductive gro
up. "Real parabolic induction" means starting with an irreducible unitary
representation pi_L of L(R)\, lifting it to P(R) by making U(R) act trivia
lly\, then applying Mackey induction from P(R) to G(R).\n\nOf course atlas
lives in the rather different world of (g\, K)-modules. I'll explain a bi
t about how to translate between these two worlds\, and what atlas can tel
l you about real parabolic induction.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220505T143000Z
DTEND:20220505T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/18
DESCRIPTION:Title: Theta-stable parabolic subgroups and cohomological induction in atla
s\nby Jeffrey Adams (University of Maryland) as part of Real reductive
groups/atlas\n\n\nAbstract\nIt has been understood since the 1950s that (
unitary) representations of reductive groups should correspond approximate
ly to (unitary) characters of Cartan subgroups. Last week we talked about
real parabolic induction\, a method also dating to the 1950s for construct
ing part of this correspondence (from the noncompact parts of Cartan subgr
oups).\n\nIn the 1970s\, Gregg Zuckerman introduced a parallel method\, ca
lled cohomological induction\, for constructing the part of the correspon
dence corresponding to compact parts of Cartan subgroups. We'll explain ho
w that works\, and how to realize it in the atlas software.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220512T143000Z
DTEND:20220512T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/19
DESCRIPTION:Title: Unitary dual of SO(2n\,1) in atlas\nby David Vogan (MIT) as part
of Real reductive groups/atlas\n\n\nAbstract\nDescription of the unitary
dual of SO(2n\,1) in atlas terms.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220519T143000Z
DTEND:20220519T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/20
DESCRIPTION:Title: Unitary dual of F4_B4 in atlas\nby David Vogan (MIT) as part of
Real reductive groups/atlas\n\n\nAbstract\nAbout the Baldoni-Silva/Barbasc
h classification of unitary representations of the rank one form of F4\, a
s seen by atlas.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220526T143000Z
DTEND:20220526T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/21
DESCRIPTION:Title: Loose ends: Hermitian representations\, more on parameters\, transla
tion and the Jantzen filtration\nby Jeffrey Adams (University of Maryl
and) as part of Real reductive groups/atlas\n\n\nAbstract\nA few miscellan
eous topics: Hermitian representations and the Hermitian dual\, more about
parameters\, translation functors. We didn't get to the Jantzen filtratio
n\; to be continued next week\n
LOCATION:https://stable.researchseminars.org/talk/atlas/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220602T143000Z
DTEND:20220602T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/22
DESCRIPTION:Title: More loose ends: translation\, Jantzen filtration\nby Jeffrey Ad
ams (University of Maryland) as part of Real reductive groups/atlas\n\nAbs
tract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220609T143000Z
DTEND:20220609T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/23
DESCRIPTION:Title: Jantzen filtration and open mic night\nby Jeffrey Adams (Univers
ity of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nI w
ill finish going over a few recent topic\, and talk about the Jantzen filt
ration. Then I'll take questions. If there's anything you'd like to discus
s this is good chance to bring it up.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:ONE WEEK BREAK
DTSTART:20220616T143000Z
DTEND:20220616T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/24
DESCRIPTION:Title: No seminar this Thursday\, resuming next week\nby ONE WEEK BREAK
as part of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220623T143000Z
DTEND:20220623T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/25
DESCRIPTION:Title: Dirac operator in atlas\nby David Vogan (MIT) as part of Real re
ductive groups/atlas\n\n\nAbstract\nThe Dirac operator was introduced by P
arthasarathy and others in the 1970s as a tool for studying unitary repres
entations. One of its most powerful aspects is Parthasarathy's Dirac inequ
ality\, which provides an upper bound on the infinitesimal character of a
unitary representation containing a particular K-type. I'll explain how to
compute this by hand\, then introduce an atlas script which does the job.
A unitary representation is said to _have Dirac cohomology_ if equality h
olds in the Dirac inequality\; such representations have been studied exte
nsively by Barbasch\, Ding\, Dong\, Huang\, Mehdi\, Pandzic\, Wong\, Ziera
u... and I apologize to those omitted. I will show how atlas computes Dira
c cohomology.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220630T143000Z
DTEND:20220630T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/26
DESCRIPTION:Title: Classifying the unitary dual (part 1 of infinitely many...)\nby
David Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nWe
've explained a lot about how atlas can offer information about individual
representations\, and certainly how it can check whether an individual re
presentation is unitary. Want to start talking about how to describe the f
ull unitary dual: to give a finite and complete description of the answers
to infinitely many questions "is it unitary?" The Dirac inequality from l
ast week is useful hint\; I will try to say how it can be part of a genera
l picture of the unitary dual\, and how it might usefully be modified.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220707T143000Z
DTEND:20220707T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/27
DESCRIPTION:Title: Affine Weyl group facets and the unitary dual\nby David Vogan (M
IT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast week we saw
that there were two kinds of finiteness problems standing in the way of a
finite description of the unitary dual. Today I'll focus on the second on
e: dividing the continuous parameters of possibly unitary representations
into a finite set of pieces where unitarity is constant. (Don't worry\, J
eff will come back soon!)\n
LOCATION:https://stable.researchseminars.org/talk/atlas/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220721T143000Z
DTEND:20220721T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/28
DESCRIPTION:Title: Vogan duality\nby Jeffrey Adams (University of Maryland) as part
of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220728T143000Z
DTEND:20220728T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/29
DESCRIPTION:Title: Arthur packets\nby Jeffrey Adams (University of Maryland) as par
t of Real reductive groups/atlas\n\n\nAbstract\nCompletion of the discussi
on of Vogan duality from last week\; application to the definition and com
putation of Arthur packets of unipotent representations.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART:20220714T143000Z
DTEND:20220714T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/30
DESCRIPTION:Title: TAKE A WEEK OFF\nby NONE as part of Real reductive groups/atlas\
n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220804T143000Z
DTEND:20220804T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/31
DESCRIPTION:Title: Duality for singular and non-integral infinitesimal character\nb
y David Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\n
Jeffrey Adams explained in the last two lectures how Vogan duality relates
representation of a real group G(R) to representations of a real form G^v
(R) of the Langlands dual group\, in the case of regular integral infinite
simal character. Today I have three goals:\n 1) to say a few words abo
ut how this representation theory duality can be rephrased as\n\n(reps of
G(R)) DUAL TO (algebraic geometry of space of Langlands parameters in ^L
G)\n\nso that it can make sense over other local fields\;\n\n 2) Expl
ain what happens for regular but NON integral infinitesimal character. (An
swer: real form of G^v is replaced by real form of pseudolevi subgroup G^v
(gamma)).\n\n 3) Explain what happens for SINGULAR infinitesimal chara
cter. (Answer: translation principle tells you everything\, but what it t
ells you is a bit complicated.)\n\nThe subgroup G^(gamma) is dual to a Lan
glands-Shelstad endoscopic group for G. What we do with G^(gamma) is certa
inly related to endoscopy\, but it is not at all precisely the same thing.
\n
LOCATION:https://stable.researchseminars.org/talk/atlas/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220811T143000Z
DTEND:20220811T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/32
DESCRIPTION:Title: Duality for singular integral infinitesimal character\nby David
Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast wee
k I repeated Jeff's description of duality as a bijection.\n\n(reps with r
egular integral infl char of real forms of G) <--->\n (reps with regul
ar integral infl char of real forms of G^\\vee)\n\nToday I will start by e
xplaining how this changes for singular integral infl char:\n\n(reps of fo
rms of G\, integral infl char singular on simple roots S) <--->\n (r
eps of forms of G^\\vee\, integral infl char\, simple roots S^\\vee in tau
invariant)\n\nCondition on the G^vee side is that the reps must be somewh
at SMALL. A special case is what Jeff discussed already\n\n(forms of G rep
s\, infl char zero) <----> (fin-diml reps for forms of G^\\vee)\n
LOCATION:https://stable.researchseminars.org/talk/atlas/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220818T143000Z
DTEND:20220818T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/33
DESCRIPTION:Title: Hermitian forms on finite- dimensional representations\nby Jeffr
ey Adams (University of Maryland) as part of Real reductive groups/atlas\n
\n\nAbstract\nThe study of signatures of Hermitian forms on finite dimensi
onal representations serves as an example of the general theory\, while ha
ving some special features\, and being surprisingly interesting in its own
right.\n\nThe finite dimensional representations of the real forms $G(\\m
athbb R)$ of $G(\\mathbb C$) are essentially independent of the real form.
However these representations are all unitary if and only if $G(\\mathbb
R)$ is compact\, and the signature of the Hermitian forms depend very much
on the real form. \n\nWe'll talk about computing this\, using an elementa
ry formula (arising from the Atlas theory of the c-form)in terms of the We
yl character formula. We'll also discuss some interesting invariants for a
finite dimensional representation: the Frobenius/Schur indicator and the
real-quaternionic indicator.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220825T143000Z
DTEND:20220825T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/34
DESCRIPTION:Title: Cohomological induction and restricting discrete series to K\nby
David Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nH
arish-Chandra's theorem (and also the Langlands classification\, which is
based on the theorem) says that a discrete series representation of a real
reductive G is specified by a character of a compact Cartan subgroup T of
G. This talk is about Zuckerman's idea of how to implement that: given a
character of a compact Cartan\, how to construct a (g.K) module. The const
ruction is most explicit as a representation of K\; so I'll talk about how
to see the restriction to K of a discrete series\, and what things we do
and don't know about that.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220901T143000Z
DTEND:20220901T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/35
DESCRIPTION:Title: More about discrete series restriction\nby David Vogan (MIT) as
part of Real reductive groups/atlas\n\n\nAbstract\nLast week I described Z
uckerman's construction of the (g\,K)-module of a discrete series represen
tation. This week I'll look at how to make that construction explicit: how
to extract the Blattner formula\n
LOCATION:https://stable.researchseminars.org/talk/atlas/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220908T143000Z
DTEND:20220908T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/36
DESCRIPTION:Title: Cohomological Arthur packets\nby Jeffrey Adams (University of Ma
ryland) as part of Real reductive groups/atlas\n\n\nAbstract\nAn important
special case of Arthur packets are those of regular integral infinitesima
l character. The trivial representation (attached to the dual principal ni
lpotent orbit) is an example. \n\nIt is known by a result of Salamanca tha
t the unitary representations with regular integral infinitesimal characte
r are precisely the cohomological representations. These are representatio
ns with non-trivial twisted $(\\mathfrak g\,K)$ cohomology. By a result of
Vogan and Zuckerman these are precisely the modules $A_\\mathfrak q(\\lam
bda)$\, constructed via cohomological induction from a unitary character o
f theta-stable Levi subgroup. \n\nThe conclusion is: assuming all is right
with the world (i.e. Arthur's conjectures) an Arthur packet consisting of
representations with regular integral infinitesimal character\nmust consi
st of certain $A_\\mathfrak q(\\lambda)$-modules. These are sometimes refe
rred to as "Adams-Johnson" packets\; these were among the first interestin
g Arthur packets to be studied in the 1980s.\n\nI'll discuss these things
in the context of Atlas.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220915T143000Z
DTEND:20220915T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/37
DESCRIPTION:Title: Cohomological Arthur Packets 2\nby Jeffrey Adams (University of
Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\n(This is a
continuation of the talk from last week)\n\nAn important special case of
Arthur packets are those of regular integral infinitesimal character. The
trivial representation (attached to the dual principal nilpotent orbit) is
an example. \n\nIt is known by a result of Salamanca that the unitary rep
resentations with regular integral infinitesimal character are precisely t
he cohomological representations. These are representations with non-trivi
al twisted $(\\mathfrak g\,K)$ cohomology. By a result of Vogan and Zucker
man these are precisely the modules $A_\\mathfrak q(\\lambda)$\, construct
ed via cohomological induction from a unitary character of theta-stable Le
vi subgroup. \n\nThe conclusion is: assuming all is right with the world (
i.e. Arthur's conjectures) an Arthur packet consisting of representations
with regular integral infinitesimal character\nmust consist of certain $A_
\\mathfrak q(\\lambda)$-modules. These are sometimes referred to as "Adams
-Johnson" packets\; these were among the first interesting Arthur packets
to be studied in the 1980s.\n\nI'll discuss these things in the context of
Atlas.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20220922T143000Z
DTEND:20220922T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/38
DESCRIPTION:Title: Cohomological Arthur packets 3/Open Mic NightI'l\nby Jeffrey Ada
ms (University of Maryland) as part of Real reductive groups/atlas\n\n\nAb
stract\nWe'll finish our discussion of cohomological Arthur packets (see t
he previous talk for an abstract). This should leave time for questions\,
about anything at all. If you have an example you'd like to see worked out
in atlas\, be prepared to ask about it.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20220929T143000Z
DTEND:20220929T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/39
DESCRIPTION:Title: Duality\, associated varieties\, and nilpotent orbits\nby David
Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nAt the b
eginning of August I talked about "duality\," which relates a category of
(g\,K)-modules to a geometric category on the Langlands dual group G^\\vee
. (I could have said "to a category of (g^\\vee\, K^\\vee)-modules" but th
e formulation above works for p-adic G as well.) Today I'll look at associ
ated varieties in this context\, recovering various notions of "duality" f
or nilpotent orbits (due originally to Spaltenstein\, Lusztig\, and Barbas
ch-Vogan).\n
LOCATION:https://stable.researchseminars.org/talk/atlas/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART:20221006T143000Z
DTEND:20221006T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/40
DESCRIPTION:Title: NO MEETING THIS WEEK!\nby NONE as part of Real reductive groups/
atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20221013T143000Z
DTEND:20221013T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/41
DESCRIPTION:Title: Duality for G reps\, nilpotent orbits\, and W reps\nby David Vog
an (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nI will try
to summarize all the kinds of duality we've talked about: for HC modules\,
for nilpotent orbits\, and for W representations. I'll try to say what th
ey have to do with each other\, and mention open problems about them.\n\nU
nderstanding all of this appropriately seems to be a way to understand ass
ociated varieties of Harish-Chandra modules.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20221020T143000Z
DTEND:20221020T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/42
DESCRIPTION:Title: Duality for G reps\, nilpotent orbits\, and W reps III\nby David
Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast we
ek I talked in some detail about duality in the case of nonintegral infini
tesimal character. This involved a pseudoLevi subgroup D^\\vee of G^vee (c
entralizer of exponential of infinitesimal character)\, and a correspondin
g endoscopic group D sharing the Cartan H of G\; roots of D are the gamma-
integral roots. \n\nDUALITY relates reps of G of infl char gamma and reps
of D^\\vee. Therefore there is an EQUIVALENCE between reps of G of infl ch
ar gamma and reps of D of (D-integral) infl char gamma). Today I'll talk a
bout how (complex) nilpotent orbits and W rep move in this equivalence. I
will try to describe a great research topic: to understand how REAL nilpot
ent orbits move between D and G.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20221027T143000Z
DTEND:20221027T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/43
DESCRIPTION:Title: Examples of Duality/Miscellaneous\nby Jeffrey Adams (University
of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nI'll do
some atlas examples illustrating the duality theory of the past few weeks
.\nThen I'll cover a few other topics which I've been asked about recently
. Possibilities include: computing derived functors outside of the good ra
nge\, details about unipotent representations of $G_2$\, large representat
ions and Whittaker models.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20221103T143000Z
DTEND:20221103T160000Z
DTSTAMP:20260404T094833Z
UID:atlas/44
DESCRIPTION:Title: More on Duality/Miscellaneous\nby Jeffrey Adams (University of M
aryland) as part of Real reductive groups/atlas\n\n\nAbstract\nThis is a c
ontinuation of last week. I'll give a few more atlas examples of duality.\
n\nLast week I defined a correspondence relating certain (special) nilpote
nt K-orbits for G to certain (special) nilpotent K^\\vee orbits for G^\\ve
e. I'll use atlas to compute some examples of this correspondence.\n\nI'll
also talk about the Speh representation\, and computin degenerate derived
functor modules in atlas.\n\nTime permitting: Arthur packets for G2.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20221110T153000Z
DTEND:20221110T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/45
DESCRIPTION:Title: Arthur packets for G2\nby Jeffrey Adams (University of Maryland)
as part of Real reductive groups/atlas\n\n\nAbstract\nWe've talked about
how to compute WEAK Arthur packets for general real reductive G: the union
over all parameters with a fixed restriction to SL(2) of the Arthur packe
t. Today I'll look at precisely how these weak packets break into actual p
ackets. This is subtle (and interesting!) and atlas cannot efficiently com
pute it for general G.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART:20221124T153000Z
DTEND:20221124T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/46
DESCRIPTION:Title: NO MEETING: US Thanksgiving Holiday\nby NONE as part of Real red
uctive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/atlas/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams\, David Vogan (MIT)
DTSTART:20221117T153000Z
DTEND:20221117T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/47
DESCRIPTION:Title: More about Arthur packets\nby Jeffrey Adams\, David Vogan (MIT)
as part of Real reductive groups/atlas\n\n\nAbstract\nBegin with Jeffrey t
alking about how a weak unipotent Arthur packet for G2 contains two non-un
ipotent Arthur packets (as constructed by Adams and Johnson in 1987). Thes
e strange overlaps account for the extra stable sums of irreducibles in th
e weak packet\, which were displayed last week.\n\nProbably David will the
n talk about this: if the SL(2) portion of an Arthur parameter psi is not
distinguished in the dual group\, then the Bala-Carter Levi L^vee for the
SL(2) is a PROPER Levi. Of course a Levi L^\\vee in G^\\vee corresponds to
a Levi L in G\, and this Levi comes with an inner class of rational forms
(although it need NOT be the Levi of a rational parabolic). The Arthur pa
rameter psi for G can be regarded as an Arthur parameter for L. In the rea
l case\, there are well-behaved "cohomological induction functors" carryin
g representations of (these rational forms of) L to representations of (ou
r inner class of) rational forms of G. \n\nTHESE INDUCTION FUNCTORS CARRY
THE ARTHUR PACKET Pi_psi(L) INTO THE ARTHUR PACKET Pi_psi(G).\n\nCONJECTUR
E: they are ONTO.\n\nIf this is true\, then Arthur packets are only diffic
ult when the SL(2) part is DISTINGUISHED\; that is\; the nilpotent is rath
er large\; that is\; the representations are rather far from tempered.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20221201T153000Z
DTEND:20221201T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/48
DESCRIPTION:Title: Computing honest Arthur packets\nby David Vogan (MIT) as part of
Real reductive groups/atlas\n\n\nAbstract\nLong general talk about Langla
nds' approach to automorphic forms\, and how it leads toward Arthur's conj
ectures. Brief demonstration of Annegret Paul's new script\, which actuall
y calculates Arthur packets in many cases.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20221208T153000Z
DTEND:20221208T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/49
DESCRIPTION:Title: Annegret Paul's magic script\nby David Vogan (MIT) as part of Re
al reductive groups/atlas\n\n\nAbstract\nA unipotent Arthur packet is atta
ched to a nilpotent class in the dual group\, and a finite amount of addit
ional data. A weak pack is the union of all packets attached to a single n
ilpotent class\; the software has been able to compute those (with great e
ffort!) for a long time. Annegret Paul has a new script which can (conject
urally) compute actual Arthur packets for those nilpotents in the dual gro
up which are principal in some Levi. Often this is most nilpotents\; for E
8\, it is 41 of the 70 nilpotent classes.\n\nI will recall the theoretical
conjecture required to PROVE this (without it\, we know only that the scr
ipt is producing _part of_ an honest Arthur packet)\; then spend most of t
he time looking carefully at how the script works. This is meant to be an
exercise both in theory and in scripting for atlas.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams\, David Vogan (University of Maryland)
DTSTART:20221215T153000Z
DTEND:20221215T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/50
DESCRIPTION:Title: Interesting examples of Arthur packets\, computed by Annegret's scri
pt\nby Jeffrey Adams\, David Vogan (University of Maryland) as part of
Real reductive groups/atlas\n\n\nAbstract\nLast time we looked a bit at t
he script "test_non_distinguished.at"\, which computes (some) Arthur packe
ts. This time will emphasize more the interesting examples\, and less the
(also interesting!) details about scripting.\n\nThis will be the LAST semi
nar of the fall semester. Probably we will not meet again at least until T
hursday\, January 11\; from that time on the schedule is still under discu
ssion.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20230112T153000Z
DTEND:20230112T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/51
DESCRIPTION:Title: Computing unitary duals\, I: cohomological induction\nby David V
ogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nWe are pl
anning to conclude this seminar (for now?) with three talks January 12\, 1
9\, and 26\, 2023. Topic is progress and plans for the original goal of th
e atlas project: to make software that can describe the unitary dual of an
y real reductive group G(R).\n\nThere are two fundamental classical techni
ques to construct unitary representations. The first (due to Israel Gelfan
d and his collaborators) is real parabolic induction. A theorem in Knapp's
"Overview" book gives a very simple way to identify most of the unitary r
epresentations that can be obtained in this way: they are the ones with no
n-real infinitesimal character. In light of that theorem\, one can study _
only_ representations with REAL infinitesimal character.\n\nThe second cla
ssical technique (due to Gregg Zuckerman and those who stole from him) is
cohomological parabolic induction. The analogue of the theorem in Knapp's
book would say that any unitary representation with non-imaginary infinite
simal character can be obtained by cohomological induction. THIS IS NOT TR
UE\, but it is nearly true.\n\nWhat's actually true is that any unitary re
presentation for which the real part of the infinitesimal character is LAR
GE ENOUGH can be obtained by cohomological induction. The question of what
"large enough" means is best expressed in terms of "nonunitarity certific
ates. Today I will state these results with some care\, and start to look
at nonunitarity certificates.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20230119T153000Z
DTEND:20230119T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/52
DESCRIPTION:Title: Computing unitary duals\, II: nonunitarity certificates\nby Davi
d Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast w
eek I stated the Langlands classification in terms of atlas parameters. I
showed how an atlas parameter defines a real parabolic P^u = M^u N^u\, and
stated the theorem from Knapp's "Overview" that the corresponding represe
ntation pi_G is unitary if and only if it's unitarily induced from pi_{M^u
}. Classifying the unitary dual therefore reduces to the case M^u = G\, wh
ich is equivalent to REAL INFINITESIMAL CHARACTER.\n\nNext\, I showed how
to attach to an atlas parameter p a theta-stable parabolic q = l+u\, and a
parameter pL for L\; and sketched how the G-rep pi attached to p is cohom
ologically induced from the L-rep piL attached to pL.\n\nTopic for today i
s to understand how the correspondence piL --> pi is related to unitarity.
\n\nApparently unrelated\, but actually deeply connected: if pi is an irre
ducible representation of a real reductive G(R)\, we get a multiplicity fu
nction m_{pi}: K^ --> N\, m_{pi}(mu) = multiplicity of mu in pi|_K. The a
tlas software computes this function (up to K-types of any specified heigh
t). If pi admits an invariant Hermitian form\, then the form has a signatu
re: m_pi is the sum of two N-valued functions p_{pi} and q_{pi}. We arrang
e the form to be positive definite on some lowest K-type mu_0\, so that p_
{pi}(mu_0) = 1 and q_{pi}(\\mu_0) = 0.\n\nA NONUNITARITY CERTIFICATE for p
i is a particular K-type mu' with the property that q_{pi}(mu') > 0.\n\nFo
r each K-type mu there is a FINITE set {mu'_0\,...\,mu'_r} of K-types so t
hat ANY NONUNITARY REP OF LKT \\mu MUST HAVE A NONUNITARITY CERTIFICATE IN
{mu'_i}. Such a set is called a NONUNITARITY CERTIFICATE FOR THE LOWEST K
-TYPE mu. I will explain how atlas can hope to compute this finite set (it
doesn't yet!) and how knowledge of such sets can help to compute unitary
duals.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20230126T153000Z
DTEND:20230126T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/53
DESCRIPTION:Title: Computing unitary duals\, III: unitary_dual@RealForm\nby David V
ogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nContent t
oday will depend on what I did or didn't manage to do on January 12 and 19
. Approximately the goal is to describe what a unitary-dual-computer in at
las might look like.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20230202T153000Z
DTEND:20230202T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/54
DESCRIPTION:Title: Examples/Open Mic\nby Jeffrey Adams (University of Maryland) as
part of Real reductive groups/atlas\n\n\nAbstract\nI will do some examples
of what David Vogan covered the last few weeks. \n\nI will leave time to
answer questions\, on anything atlas-related. Feel free to ask about anyth
ing from issues using the software\, to research level questions. I expect
to be able\nto answer most questions of the first type\, and maybe think
of an interesting atlas example\nto address the second.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20230209T153000Z
DTEND:20230209T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/55
DESCRIPTION:Title: Question and Answer session\nby Jeffrey Adams (University of Mar
yland) as part of Real reductive groups/atlas\n\n\nAbstract\nThe seminar p
roper has concluded for the time being. However I'll be available to answe
r questions about the software\, the math behind it\, and whether pigs hav
e wings.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART:20230216T153000Z
DTEND:20230216T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/56
DESCRIPTION:Title: question/answer session\nby David Vogan (MIT) as part of Real re
ductive groups/atlas\n\n\nAbstract\nQuestions mostly concerned interpretin
g atlas K-types and inducing data for representations. The "slides" link i
s transcript of the atlas session.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART:20230223T153000Z
DTEND:20230223T170000Z
DTSTAMP:20260404T094833Z
UID:atlas/57
DESCRIPTION:Title: Q&A on linear algebra in atlas\nby Jeffrey Adams (University of
Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nJeff discu
ssed solving linear equations in atlas\, computing weight multiplicities\,
and some related mathematics.\n
LOCATION:https://stable.researchseminars.org/talk/atlas/57/
END:VEVENT
END:VCALENDAR