RTG Representation Theory Seminar
This is a learning seminar on topics in representation theory organized by Charlotte Chan and Karol Koziol. 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 2021
 1/24 Planning meeting
 1/31 Trace formula for SL2 and its stabilization [speaker: Tasho Kaletha] ** 2 hour talk: 3p5p **
 2/7
 2/14
 2/21
 3/7
 3/14
 3/21
 3/28
 4/4
 4/11