Representations of Reductive Groups
Speakers, Titles, Abstracts
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| Pramod Achar (LSU) | Dan Barbasch (Cornell) | ||
| Leticia Barchini (Oklahoma State) | Paul Baum (Penn State) | ||
| Dan Ciubotaru (Utah) | Meinolf Geck (Stuttgart) | ||
| Tasho Kaletha (Princeton) | Toshi Kobayashi (Tokyo) | ||
| Bertram Kostant (MIT) | George Lusztig (MIT) | ||
| Colette Moeglin (Paris) | Allen Moy (HKUST) | ||
| Kyo Nishiyama (Aoyama Gakuin) | Eric Opdam (Amsterdam) | ||
| Mark Reeder (Boston College) | Gordan Savin (Utah) | ||
| Wilfried Schmid (Harvard) | David Vogan (MIT) | ||
| Jean-Loup Waldspurger (Paris) | Nolan Wallach (San Diego) | ||
Pramod Achar, Parity sheaves on the affine Grassmannian and the
Mirkovic--Vilonen conjecture
Let G be a connected complex reductive group, and let Gr denote its
affine Grassmannian. This space has the remarkable property that its
topology encodes the representation theory of the split Langlands
dual group G^\vee over any field k. More precisely, the geometric
Satake equivalence, in the form due to Mirkovic--Vilonen, says, in
part, that certain intersection cohomology groups on Gr with
coefficients in k realize the irreducible representations of G^\vee
over k. This result raises the possibility of using the universal
coefficient theorem of topology to compare representations over
different fields. With that in mind, Mirkovic and Vilonen
conjectured in the late 1990's that the local intersection cohomology
of the affine Grassmannian with integer coefficients is torsion-free.
I will try to explain these ideas with examples, and I will discuss
the recent proof of (a slight modification of) the Mirkovic--Vilonen
conjecture. This is joint work with Laura Rider.
Dan Barbasch, Hermitian forms
for Iwahori-Hecke algebras
This talk will discuss star operations for Iwahori-Hecke algebras,
joint work with Dan Ciubotaru. Hecke algebras are structures which are
used to study the representation theory of p-adic groups. In
particular by results of Barbasch- Moy and subsequently
Barbasch-Ciubotaru, there is a precise relation between the unitary
dual of a component of representations of a p-adic group (in the sense
of Bernstein) and a particular Iwahori-Hecke algebra. In order to talk
about unitarity for an algebra, one needs a star operation. For
semisimple Lie algebras, star oper- ations are essentially
parametrized by real forms. An analogous situation exists for Hecke
algebras, but the situation is more rigid, and these new star
operations do not seem to have a natural interpretation in the larger
context of a p-adic group.
Interpretations of these star operations in the context of the representation the- ory of p-adic groups will be discussed.
Some of this work is modeled on the one hand by results in the real case by Adams-Trapa-vanLeuwen and Vogan, on the other by work of Schmid-Vilonen. 1
Leticia Barchini, Two triangularity results and invariants
of (\sp(p,q),Sp(p)xSp(q)) modules
Click here for a pdf file of the abstract.
Paul Baum, Geometric structure and the local Langlands correspondence
Let G be a connected split reductive p-adic group. Examples of connected
split reductive p-adic groups are
GL(n, F) , SL(n, F), SO(n, F), Sp(2n, F), PGL(n, F), PSL(n, F) where n can
be any positive integer and F can
be any finite extension of the field Q_p of p-adic numbers. The smooth dual
of G is the set of (equivalence
classes of) smooth irreducible representations of G. The representations are
on vector spaces over the complex
numbers. In a canonical way, the smooth dual of G is the disjoint union of
countably many subsets known as the
Bernstein components. Calculations in K-theory and periodic cyclic homology
indicate that a very simple geometric
structure might be present in the smooth dual of G. The ABPS
(Aubert-Baum-Plymen-Solleveld) conjecture makes
this precise by asserting that each Bernstein component in the smooth dual
of G is a complex affine variety. These
varieties are explicitly identified as certain extended quotients. Granted a
mild restriction on the residual characteristic
of the p-adic field F over which G is defined, the ABPS conjecture has
recently been proved for any Bernstein component
in the principal series of G. A corollary is that the local Langlands
conjecture is valid throughout the principal series of G.
What happens with connected reductive p-adic groups which are not split?
Some comments will be made about this issue.
The above is joint work with Anne-Marie Aubert, Roger Plymen, and Maarten
Solleveld.
Dan Ciubotaru, Spin representations and Green polynomials of Weyl groups
I will present a new realization of the irreducible characters of
the pin double cover of Weyl groups, obtained in joint work with Xuhua He.
The method combines elements of the Lusztig-Shoji algorithm for the
calculation of graded Springer representations with the theory of the Dirac
operator for graded Hecke algebras (previously introduced with D. Barbasch
and P. Trapa).
Meinolf Geck, Verifying Kottwitz's conjecture by computer, Part II
Let W be a finite Weyl group. Kottwitz (2000) introduced
certain involution modules for W and conjectured that they
give precise information about the intersections of left
cells with conjugacy classes of involutions. This talk
refers to an article by Casselman with the same title, in
which parts of Kottwitz' conjecture were verified for some
groups of exceptional type. We shall present work of
A. Halls (Aberdeen) and explain the theoretical and
algorithmic methods which were needed to complete the
verification for the hardest case: type E_8.
Tasho Kaletha, Rigid inner forms and endoscopy
Adams, Barbasch, and Vogan defined the notion of a strong real form of a
connected reductive group defined over the real numbers. It is a
rigidification of the usual notion of an inner form and Vogan posed the
problem of finding an analogous notion in the p-adic case. We will describe
a new cohomology set for affine algebraic groups defined over local fields
of characteristic zero. Its construction is uniform for real and p-adic
groups and it leads to the notion of a rigid inner form which in the real
case turns out to be equivalent to that of a strong real form. We will
discuss applications to the internal structure of L-packets and the
stabilization of the Arthur-Selberg trace formula. Afterwards, we will
discuss examples of constructions of L-packets for p-adic groups whose
internal structure and endoscopy fit the proposed conjectural description.
These include the depth-zero supercuspidal representations considered in
the
work of DeBacker and Reeder, as well as the epipelagic representations
defined and studied in the recent work of Reeder and Yu.
Toshi Kobayashi, Multiplicities in the restriction and real spherical varieties
Branching problems ask how irreducible representations of groups
decompose when restricted to subgroups. Decompositions of tensor product
representations, Littlewood-Richardson's rules, and Blattner formulae are
classical examples of branching laws for symmetric pairs. However, we
observe that bad features like "infinite multiplicites" may well happen in
dealing with branching problems of irreducible representations of real
reductive groups G when restricted to maximal reductive subgroups G', even
if (G,G') are symmetric pairs. In this talk I plan to discuss what is a
"nice framework" in which we could ecpect to develop a fruitful and detailed
analysis on branching laws. In connection with the theory of "real spherical
varieties", I plan to give a classification of reductive symmetric pairs
(G,G') for which multiplicites are always finite/bounded, respectively.
Bertram Kostant, On the singular elements of a semisimple Lie
algebra and the generalized Amitsur-Levitski theorem
I connect an old result of mine on a Lie algebra generalization of the
Amitsur-Levitski theorem with recent results of Kostant-Wallach on the
variety of singular elements in a reductive Lie algebra.
George Lusztig, On conjugacy classes in reductive groups
In this talk we will study a correspondence between the
conjugacy classes in a reductive group and the conjugacy classes
in the Weyl group.
Colette Moeglin, Elliptic representations in the twisted case
Allen Moy, On cosets of depth zero lattices
We present an example of a rigidity property for cosets of
depth zero lattices in the Lie algebra sl(2), and an application via,
the Fourier Transform, to distributions supported on the topological
nilpotent set. This investigation is joint with Fiona Murnaghan and
Xuhua He. Extrapolation of this property to higher rank would be
interesting.
Kyo Nishiyama, On K-spherical flag varieties
We study K-orbits on double flag varieties for symmetric pair G/K,
and deduce some useful criterions to detect the finiteness of K-orbits.
Some of them work completely for the double flags where one of the flags is
complete. We give complete classifications of such cases, which in turn
implies a classification of K-spherical flag varieties. The talk is based
on a joint work with Xuhua He, Hiroyuki Ochiai and Yoshiki Oshima.
Eric Opdam, On a remarkable uniqueness property of
cuspidal unipotent representations of unramified p-adic groups
A result of Mark Reeder states that for exceptional split
groups over a non-archimedean local field, there exists a unique (up to the
action of the group of rational characters) partitioning of the unipotent
discrete series in packets of representations whose formal degrees share the
same q-rational factor. We will show in this talk that for any unramified
group the q-rational part of the formal degree of a cuspidal unipotent
representation determines a unique unramified Langlands parameter, up to the
action of the group of rational characters. The proof uses existence of
certain so-called spectral transfer maps between affine Hecke algebras and,
in turn, the result implies a uniqueness property for such spectral transfer
maps between affine Hecke algebras.
This is joint work with Yonqi Feng.
Mark Reeder, Depths of representations of p-adic groups
A reductive p-adic group G is like an ocean, whose surface is the
building of G. At each point x on the surface, Moy and Prasad have
defined a canonical descending sequence of compact open subgroups of G
whose higher quotients V(x,r), for r>0, are abelian and afford
representations of the top quotient G(x).
Moy and Prasad showed that each irreducible admissible representation of G
contains a character of some V(x,r)
arising from a semistable orbit of G(x) in the dual of V(x,r), in the sense of
Geometric Invariant Theory.
The problem then arose to classify the pairs (x,r) for which V(x,r) has
semistable orbits.
I will describe the classification of these semistable orbits in Moy-Prasad filtrations. This is joint work with Jiu-Kang Yu.
Gordan Savin, Problems in the theory of invariant polynomials arising from theta
correspondences
Rings of invariant polynomials appear naturally in the work of Gross,
Reeder, and Yu on representations of small positive depth. Computing theta
correspondence explicitly for these representations leads to certain
problems in the theory of invariant polynomials, which can be formulated
over any field.
Wilfried Schmid,
The \n-cohomology of the limits of the discrete series.
Work of Carayol, recently elaborated by Grifiths, Green, and
Kerr, has revived interest in the $\mathfrak n$-cohomology of the discrete
series. Soergel has given a somewhat inexplicit description in a 1997
paper, but his argument contains a gap that he has been unable to close.
After discussing the relevance of the problem I shall give an answer and
outline its proof.
David Vogan, Relations among different Hecke algebra modules and nilpotent orbits
In a 1983 paper, Lusztig and I introduced a Hecke algebra module with
a basis indexed by equivariant local systems for a symmetric subgroup
acting on a flag variety. To each such local system x corresponds a
twisted involution theta(x) for the Weyl group (W,S).
In a 2012 paper, Lusztig and I introduced a Hecke algebra module with a basis indexed by twisted involutions theta for (W,S). It is natural to expect some relationship between the old modules and the new, but I do not know how to write one down. Nevertheless there are shadows of such a relationship that _can_ be written down, and which are related in turn to the ideas of Kottwitz appearing in the lecture of Meinolf Geck. I will discuss those ideas; nilpotent coadjoint orbits will appear because I can't say two words without mentioning them.
J.L. Waldspurger, Some results on twisted endoscopy for real groups.
To stabilize the twisted Arthur-Selberg's trace formula, some
preliminary results are needed. In particular, we must dene spectral
transfer. In the non-twisted case and over a non-archimedean local
eld, it is the subject of the paper of Arthur published in Selecta
Math. Over the real field, these results come from the work of Shelstad
in the non-twisted case and probably from the recent work of Mezo in
the twisted case. We present another proof, over the real field and in
the twisted case, closer to the paper of Arthur.
Nolan Wallach, The condition of moderate growth
First we will explain the version of the "Casselman-Wallach
Theorem" that is in my book Real Reductive Groups II. The second part will
be a discussion of holomorphic families of admissible finitely generated
(g,K)-modules (in particular, discussing some results of V. van der
Noort) and the extent to which the moderate growth Fréchet completions of
the modules vary in a weakly holomorphic manner.