MIT Lie Groups Seminar

Abstract: The notion of endoscopic group was created by Langlands, Shelstad, and others beginning in the 1970s, in order to study problems in harmonic analysis on a reductive group $G$: for example, the nature of {\em characters} of irreducible representations of $G$. An endoscopic group $H$ for $G$ is a smaller reductive group, equipped with a natural ``endoscopic transfer'' map from characters of $H$ to characters of $G$.

Perhaps the simplest example of such an endoscopic group is the Levi subgroup $M$ of a rational parabolic subgroup $P = MU$. Endoscopic transfer in this case is just the Mackey-Gelfand notion of {\em parabolic induction} from $M$ to $G$.

This example of rational parabolic induction is almost never mentioned in the literature on endoscopy, because endoscopy offers little that is new in that case. But I believe that it sheds some light on the nature of the Langlands-Shelstad theory.

I will talk about a (real groups) generalization of endoscopic groups and endoscopic transfer, for which the simplest example is the Levi subgroup $L$ of a theta-stable parabolic; and transfer is {\em cohomological induction} from $L$ to $G$. The formalism appears to make sense for any local field, and should lead to a generalized definition of endoscopic transfer once an appropriate local Langlands conjecture is proved.

This is joint work with Lucas Mason-Brown and Jeffrey Adams.

Slides

Video

Abstract: Let G be a connected reductive group over an algebraically closed field. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. In this talk, we shall first recall the construction of Chevalley group schemes by Lusztig using quantum groups. Then we shall discuss the construction of symmetric subgroup schemes parameterising symmetric subgroups K of G using quantum symmetric pairs. The existence of such group schemes allows us to apply characteristic p methods to study the geometry of K-orbits on the flag variety of G. This leads to a construction of Frobenius splittings via quantum symmetric pairs, generalising the algebraic Frobenius splittings by Kumar-Littelmann. This is based on joint work with Jinfeng Song (NUS).

Video

Abstract: We define pre-cuspidal families in proper parabolic subgroups of a Weyl group and show how to use them to index the irreducible representations of that Weyl group in terms of certain pairs of finite groups.

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