NUS Representation Theory Seminar

Schedule of Talks

• Thursday, 3 pm, Sept. 4 S17 05-11: Max Gurevich (Technion I.I.T.), Distributional methods on the $$p$$-adic mirabolic group.

Abstract:
Given a $$p$$-adic group $$G$$ and its homogeneous space $$X=G/H$$, the irreducible sub-representations of the regular $$G$$-action on $$C^\infty(X)$$ are a focus of much interest (part of the so-called relative harmonic analysis), sometimes motivated by periods on automorphic forms. The study of distributions spaces on $$X$$, or on its tangent space, is a central tool for this theory. On a different note, in the case of $$G=GL_n(F)$$ ($$F$$ $$p$$-adic), the mirabolic subgroup $$P\subset G$$, that is, the group of matrices whose bottom row is $$(0\ldots 01)$$, is known to play a special role in the representation theory of $$G$$. I will show some evidence which suggests that $$P$$ may retain this role in the relative setting as well. In particular, I will overview results on some symmetric spaces $$X= GL_n(F)/H$$ that are analogous to Bernstein's result on $$P$$-invariant distributions for the group case. These incorporate new distributional techniques.
• Thursday, 3 pm, Sept. 11 S17 05-11: XIONG Wei (NUS), Explicit Waldspurger formula and non-congruent numbers.

Abstract: A positive integer is called a congruent number if it occurs as the area of a right triangle whose sides have rational length, otherwise it is called non-congruent. Recently, Ye Tian, Xinyi Yuan and Shou-Wu Zhang found new criterions for non-congruent numbers. A key ingredient in their proof is an explicit version of Waldspurger's period formula. In this talk, I will explain their result and the method of proof for a square-free integer congruent to 1 modulo 8.
• Thursday, 3 pm, Oct. 2 S17 05-11: ZHANG Lei (NUS), Whittaker functions on metaplectic groups and quantum groups.

Abstract: We will discuss Whittaker functions on covers of reductive groups and will give an explicit formula for certain Whittaker coefficients in terms of crystal graphs. Such a formula provides a web of connections between automorphic forms and Weyl group multi-variable Dirichlet series, partition functions from statistical mechanics and Schubert varieties.
In this talk, we will focus on the connection to Lusztig data for canonical bases on the dual group using a result of Kamnitzer, which works for general split reductive groups.
Moreover, as an application of our results, we obtain a generalization of Tokuyama formula in the case of the Lie algebra $$B_n$$.
• Thursday, 3:30 pm, Oct. 16 S17 04-01: Martin Weissman (Yale-NUS), Central extensions of reductive groups by K1 and K2.

Abstract: This lecture will cover the classification of central extensions of reductive groups by K1 and by K2 (algebraic K-theory sheaves). Since the K-theory sheaf K1 is represented by the multiplicative group GL(1), the first case is an easy but important warm-up involving root data. After the K1 case, I will focus on the central extensions of G by K2 -- the Brylinski-Deligne extensions which have become widely utilized in the study of covering groups. The second half of the lecture will serve as an introduction to Brylinski-Deligne extensions over fields, and extensions of the Brylinski-Deligne classification to reductive groups over certain rings of integers. This paves the way for a study of Bruhat-Tits theory for Brylinski-Deligne extensions. I will also discuss examples of covers of semisimple Lie groups.
• Thursday, 3:30 pm, Oct. 23 S17 04-01: Martin Weissman (Yale-NUS), L-gerbes and L-groups for covering groups.

Abstract: The previous lecture introduced the classification of Brylinski-Deligne extensions of reductive groups by K2. This classification involves a triple of invariants (Q,D,f). In this lecture, I will use this data to construct an L-group associated to a quasisplit group G over a local or global field F, a Brylinski-Deligne extension G' of G, and a positive integer n for which F has n distinct nth roots of unity. I will focus on cases in which representations of a covering group can be naturally parameterized by Weil parameters valued in this L-group.
• Thursday, 3 pm, Nov. 6 S17 05-11: GAO Fan (NUS), Exercises on Brylinski-Deligne extensions: covers of $${\rm SL}_2$$ and $${\rm PGL}_2$$.

Abstract: To supplement the previous two talks by M. Weissman, we work out certain simple but instructive examples of Brylinski-Deligne covers for $${\rm SL}_2$$ and $${\rm PGL}_2$$ in details. In particular, we give concrete descriptions for the covers and their associated L-groups. Properties of the L-groups will be discussed, from which we see that interesting phenomenon arise.
• Tuesday, 3 pm, Dec. 9 S17 05-11: Michaël Pevzner (University of Reims), Differential Symmetry breaking operators for reductive pairs.

Abstract: Rankin--Cohen brackets are the symmetry breaking operators for the tensor product of two particular representations of $${\rm SL}(2,\mathbb{R})$$. We discuss the general problem to find explicit formulae for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs.
Our approach is based on an algebraic Fourier transform for generalized Verma modules, which enables us to characterize those differential symmetry breaking operators by means of Gauss hypergeometric functions.
We will present some explicit formulae, of new equivariant holomorphic differential operators for all the six different complex geometries arising from real symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of Jacobi polynomials appear in these operators including the classical Rankin--Cohen brackets as a special case.