RTG Number Theory Seminar, Fall 2022: the SakellaridisVenkatesh conjectures
This is a learning seminar on the relative Langlands program organized by Tasho Kaletha, Kartik Prasanna, and Charlotte Chan. The focus of the seminar will be on the SakellaridisVenkatesh conjectures. We meet 3:004:15p on Mondays in EH 4088. Talks should run around 60 minutes, with the remaining time left for questions, comments, discussion, and chatting. (Note that GLNT will meet 4:30p5:30p on Mondays, also in EH 4088.)
Literature:
 [SK] Sakellaridis and Venkatesh, Periods and harmonic analysis on spherical varieties
 [L] Luna, Variétés sphériques de type A
 [BP] Bravi and Pezzini, Primitive wonderful varieties
 [LV] Luna and Vust, Plongements d'espaces homogènes
 [K] Knop, The LunaVust theory of spherical embeddings
 [KS] Knop and Schalke, The dual group of a spherical variety
 [S] Sakellaridis, Spherical functions on spherical varieties
 [GG] Gan and Gomez, A conjecture of SakellaridisVenkatesh on the unitary spectrum of spherical varieties
Lecture notes for all talks [by Guanjie Huang]
Schedule: 9/12 Overview on the absolute and relative Langlands programs + planning meeting [speaker: Tasho Kaletha]
 9/19 Structure theory of homogeneous spherical varieties [L, BP] [speaker: Robert Cass]
 9/26 LunaVust theory: structure theory of spherical embeddings (nonhomogeneous spherical varieties) [LV, K] [speaker: Calvin YostWolff]
 10/3 The dual group of a spherical variety (after KnopSchalke) [KS] [speaker: Charlotte Chan]
 10/10 Local geometry and asymptotics [SV, §4,5] [speaker: Alex Bauman]
 10/24 Bernstein morphisms, scattering theory, and the Plancharel formula: an overview [SV, §915] [speaker: Guanjie Huang]
 10/31 The SakellaridisVenkatesh conjectures [SV, §1618] [speaker: Kartik Prasanna]
 11/7 Spherical functions and Lfunctions [S] [speaker: Jialiang Zou]
 11/14 Local relative character for some strongly tempered spherical varieties [speaker: Chen Wan]
 11/21 Overview of the theta correspondence [speaker: Lukas Scheiwiller]
 11/28 The work of GanGomez [speaker: Guanjie Huang]
 12/5 Introduction to the relative trace formula [speaker: Elad Zelingher]
Abstract: We present Luna's classification of homogeneous spherical varieties for a reductive group G, i.e., spherical varieties of the form G/H. The classification problem reduces to understanding those subgroups H which are "spherically closed." In this case, G/H has a unique wonderful compactification, and the problem is to classify wonderful varieties by combinatorial data. Luna achieved this classification for groups of type A, and much later Bravi and Pezzini showed Luna's classification works for general groups. Along the way we will give examples of spherical and wonderful varieties, and we will encounter the important notion of spherical roots of G.
Abstract: The classifications of spherical embeddings characterizes spherical varieties with dense G orbit G/H according to certain combinatorial data called colored fans. I will present the classification of spherical embeddings along with some examples. Then I will show how the colored fan relates to the geometry of the spherical variety along with how morphisms of spherical embeddings correspond to maps of colored fans.
Abstract: We explain KnopSchalke's construction of the dual group of any Gvariety X. Their approach is based on first associating to X a "weak spherical datum", which is a weakening of Luna's homogeneous spherical datum. To this weak spherical datum, one associates two dual groups: one using weak spherical roots (this is the dual group of X) and one using associated roots (this is a subgroup of the dual of G). The construction of the SakellaridisVenkatesh's desired distinguished morphism is then obtained by analyzing centralizers of maps between these two dual groups. We mention some functoriality properties of KnopSchalke's construction, which will likely come up later in SakellaridisVenkatesh's conjectures on "boundary degenerations" of spherical varieties.
Abstract: We construct certain "boundary degenerations" X_\Theta of a Gspherical variety X associated to a set \Theta of simple spherical roots of X, which are spherical varieties which look similar to X in some ways, but which are simpler and have extra automorphisms. We follow SakellaridisVenkatesh's study of the local geometry and asymptotics of X over a local field. First, they construct a bijection, for each compact open subgroup J, between J orbits on X and on X_\Theta near infinity. Then, they construct a Gequivariant map between the smooth functions on these spaces. The purpose of these maps are to break up the space of functions on X in terms of the boundary degenerations, which are simpler.
Abstract: The study of the Plancherel decomposition of L^2 space of a spherical variety X is the core part of SakellaridisVenkatesh paper. In this talk, we will see how this can be reduced to the discrete spectrum of its boundary degenerations. In particular, we will construct the Bernstein morphisms which span L^2(X) from the discrete spectra of its boundary degenerations, and introduce the scattering theory to describe the possible overlap. If time permits, we will talk about how to write down explicit formulas for these morphisms, and how they lead to an explicit Plancherel decomposition of L^2(X).
Abstract: The goal of the SV conjectures is to understand the relation between automorphic periods and Lvalues, as well as questions about distinction, both locally and globally. The motivating theorem is that of TunnellSaitoWaldspurger (TSW) for GL_2 and its inner forms, later generalized by the GanGrossPrasad (GGP) and IchinoIkeda (II) conjectures. The SV conjectures are a further vast generalization of this circle of ideas to the setting of spherical varieties, though not yet formulated at the same level of precision. I will start by recalling the work of TSW, GGP and II to put things in context, then explain how the SV conjectures generalize all of this.
Abstract: In this talk, we study the unramified spectrum of a homogeneous spherical variety X . We will discuss Sakellaridisâ€™s work on computing the eigenfunctions on spherical varieties under the action of the spherical Hecke algebra, which generalise the classical Casselman Shalika type formula. We will also discuss a variant of this formula, which involves the dual group of the spherical variety X and certain quotient of Lfunctions. As an application, we will present the unramified Plancherel formula for X.
Abstract: In this talk, I will explain how to compute the local relative character for strongly tempered spherical varieties in the unramified case. I will first explain the general strategy of the computation, then I will give some specific examples. This is a joint work with Lei Zhang.
           
RTG Representation Theory Seminar, 20212022: Everything SL2
This is a learning seminar on topics in representation theory organized by Karol Koziol and Charlotte Chan. We meet 4:005:15p on Mondays in EH 4088. All talks should be focused on SL2 or GL2, in varying fonts. Talks should run around 60 minutes, with the remaining time left for questions, comments, discussions, and chatting.
Please email Karol Koziol (kkoziol [at] umich [dot] edu) to get on the mailing list.
Fall 2021
 9/13 Planning meeting

9/20 Highest weight theory [speaker: Alex Bauman] [notes]
Abstract: In this talk, we introduce the Lie algebra sl_2(C) and find all its finite dimensional irreducible representations. We then show that every finite dimensional representation is a direct sum of irreducible representations. Finally, we introduce highest weight modules in general for sl_2(C) and discuss the Verma modules.

9/27 BorelWeilBott over C [speaker: Calvin YostWolff] [notes by Guanjie Huang]
Abstract: The BorelWeilBott theorem describes the cohomology of line bundles on flag varieties as certain representations. In particular, the BorelWeilBott theorem gives a geometric construction of the finite dimensional irreducible representations for reductive groups. In this talk, I will explicitly compute these representations for SL_2(C). I will then motivate our previous computations with induced representations and Serre duality, leading to the BorelWeilBott theorem for SL_2(C). Lastly, I will use the AtiyahBott fixed point formula to deduce the Weyl character formula from our geometric representations.

10/4 DeligneLusztig varieties [speaker: Andy Gordon] [notes by Andy Gordon]
Abstract: In this talk we will construct the irreducible characteristic 0 representations of $SL_2(\mathbb{F}_q)$ for $q$ a prime power. This will be done via introducing the Drinfeld curve, a smooth affine variety whose cohomology groups contain the irreducible representations.

10/11 Weil representations and irreducible representations of SL2(Fq) [speaker: Calvin YostWolff] [notes by Guanjie Huang]
Abstract: The Weil representation gives an algebraic method to classify the irreducible representations of symplectic groups Sp(2n,Fq). In this talk, we will work out this method on Sp(2,Fq) = SL(2,Fq), and show how in this case, it associates to each irreducible representation of SL(2,Fq) a maximal torus. We will first construct the Heisenberg group corresponding to a symplectic vector space and classify its irreducible representations via the finite StoneVonNeumann theorem. We then use an action of Sp(4,Fq) on the irreducible representations to construct the Weil representation of Sp(4,Fq). Finally, we will decompose the Weil representation into irreducible representations of Sp(2,Fq) via Howe duality.

10/25 Quantum sl2 and knots, I [speaker: Ilia Nekrasov]

11/1 Quantum sl2 and knots, II [speaker: Linh Truong]
Abstract: I will give an introduction to knots, the Jones polynomial, ribbon categories, and quantum knot invariants. We will focus on the quantum sl_2 knot invariant, which categorifies the Jones polynomial.

11/15 Representations of SL2(padic) [speaker: Guanjie Huang] [notes by Guanjie Huang]
Abstract: The supercuspidal representations of a padic group are those which cannot be obtained by parabolic induction. The usual way to construct a supercuspidal representation is compact induction. In this talk, we will construct the supercuspidal representations of SL(2,F) for a padic field F following Yu's construction. We will first explain how to obtain the socalled cuspidal Gdata by studying the structure of SL(2,F). Then we will use these data to construct supercuspidal representations of SL(2,F) of zero and positive depth.

11/29 Representations of SL2(R) [speaker: Havi Ellers, notes by Havi Ellers]
Abstract: There are two parts to this talk, which are for the most part disjoint from each other. In the first part we introduce a diagramatic representation of indecomposable, quasisimple, hmultiplicity free sl(2)modules. In the second we discuss the vanishing of matrix coefficients for certain representations of SL2(R).

12/6 Dynamics and SL2(R) [speaker: Carsten Peterson]
Abstract: PSL(2, R) acts simply transitively on the unit tangent bundle of the hyperbolic plane. Under this identification, we can identify the left action of PSL(2, R) on itself as Mobius transformations. As for the right action, the diagonal subgroup corresponds to the geodesic flow, and the upper triangular subgroup corresponds to the horocycle flow. These actions descend to Gamma\PSL(2, R) which we may identify with the unit tangent bundle of a hyperbolic surface. When the hyperbolic surface has finite volume, since PSL(2, R) preserves Haar measure, we get that PSL(2, R) acts by measurepreserving transformations, and hence we get a unitary representation. Ergodicity of the geodesic flow corresponds to this representation, when restricted to the diagonal subgroup, only having one copy of the trivial representation, and mixing of the geodesic flow corresponds to vanishing of matrix coefficients at infinity in the orthogonal complement of the constant functions. Both of these hold true on hyperbolic surfaces of finite volume because of the HoweMoore theorem. Nontrivial irreducible unitary representations of PSL(2, R) can be classified as either principal series, complementary series, or discrete series. SO(2) invariant functions on Gamma\PSL(2, R) correspond to functions on the hyperbolic surface. The Casimir operator restricted to such functions acts the same as the Laplacian on the hyperbolic surface. When the hyperbolic surface is compact, L^2(Gamma\PSL(2, R)) decomposes as a direct sum of irreducible representations. The complementary series in this decomposition are parametrized by eigenvalues of the Laplacian in (0, 1/4), and the principal series are parametrized by the eigenvalues in [1/4, infty) (and the trivial representation corresponds to the eigenvalue 0).
Winter 2022
 1/24 Planning meeting

1/31 Trace formula for SL2 and its stabilization [speaker: Tasho Kaletha] ** 2 hour talk: 3p5p ** **On Zoom: https://umich.zoom.us/j/96092436197, login required**

2/7 The Steinberg representation of G [speaker: Karthik Ganapathy]
Abstract: An algebraic group G is geometrically reductive if for every nonzero Ginvariant vector v in a Grepresentation V, there exists a nonconstant homogeneous Ginvariant polynomial function on V which does not vanish on v. We will see Haboush's proof that every reductive group is geometrically reductive. The main character of this proof is the Steinberg representation; the bulk of the talk will be about this representation.

2/14 Characteristic p representations of SL2Fp [speaker: Nate Harman]

2/21 Modular supercuspidal representations of SL2Qp [speaker: Karol Koziol]

3/7 TunnellSaito: local epsilon factors and representations of GL2 [speaker: Alex Bauman]

3/14 Henniart's characterization of supercuspidal representations of GL2(nonarch) [speaker: Charlotte Chan]
Abstract: In the 1990s, Henniart proved that the (complex) supercuspidal representations of GL2(nonarch local field) can be distinguished by their character on a special set of regular semisimple elements which he called "very regular." Although the character formula of these representations is quite complicated in general, the character formula on these very regular elements is very simple: up to a sign, it is equal to the average over the Worbit of an admissible character. This has seen many applications over the years, as it allows one to recognize a supercuspidal representation from a very small amount of simple data.

3/21 Representations of parahoric subgroups [speaker: Guanjie Huang]
Abstract: In this talk I will introduce Lusztig's cohomological construction of representations of reductive groups over finite rings, and compute the representations one can obtain in this way of SL2 in equal characteristic and rank 2. Lastly, I will introduce how this construction can be generalized to parahoric subgroups.

3/28 Satake isomorphism for SL2 [speaker: Carsten Peterson]
Abstract: The HarishChandra isomorphism allows one to understand the structure of the center of the universal enveloping algebra of a semisimple Lie algebra. It is also useful in understanding the structure of the algebra of invariant differential operators on a symmetric space. The Satake isomorphism is somewhat of a padic analogue which allows one to understand the structure of the spherical Hecke algebra of a semisimple padic Lie group. Special attention will be paid to how these isomorphisms work for SL_2.

4/4 Geometric Satake for SL2 [speaker: Andy Gordon] [notes by Andy Gordon]

4/11 Character sheaves for SL2 [speaker: Calvin YostWolff]

4/18 Finite subgroups of SL2C [speaker: Henry Talbott] [slides by Henry Talbott]