## RESTRICTION OF GENERALIZED VERMA MODULES TO SYMMETRIC PAIRS

**Speaker(s): ** Dr.Haian He (BICMR, PKU)

**Time: ** 00:00-00:00 December 30, 2014

**Venue: ** Room 77201 at #78 courtyard, Beijing International Center for Mathematical Research

Speaker: Dr.Haian He (BICMR, PKU)

Time: 13:45--14:45, 30 December

Abstract: Let \mathfrak{g} be a complex semi-simple Lie algebra with a parabolic subalgebra \mathfrak{p}. A module X of \mathfrak{g} is called discretely decomposable if there exists a filtration of \mathfrak{g}-modules of finite length, the union of which is X. Now let \tau be an involutive automorphism of \mathfrak{g} with the set of fixed point \mathfrak{g}'. Moreover, denote by G the adjoint group Int(\mathfrak{g}) of \mathfrak{g}, and G' and P the corresponding subgroup of \mathfrak{g}' and \mathfrak{p} respectively. The main result is that the followings are equivalent: 1) G'P is closed in G; 2) For any simple \mathfrak{g}-module X in the generalized BGG category O^\mathfrak{p}, the restriction of X to \mathfrak{g}' contains at least one simple \mathfrak{g}'-module. 3) For any simple \mathfrak{g}-module X in the generalized BGG category O^\mathfrak{p}, the restriction of X to \mathfrak{g}' is discretely decomposable as a \mathfrak{g}'-module. This job is due to KOBAYASHI Toshiyuki (小林 俊行 さん). In this talk, I shall assume that you already fully understand finite dimensional Lie algebra with its representation theory, and generalized BGG category. Also, I shall assume that you are familiar with the definition and some fundamental properties of linear algebraic group and Lie group.

Place: Room 77201 at #78 courtyard, Beijing International Center for Mathematical Research