Representation Theory seminar

Wednesday, 13 September 2023, 9:30 am

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Venue: Mini Conference Room

Host: U. K. Anandavardhanan 

Speaker: Mohammed Saad Qadri

Affiliation: IIT Bombay

Title: On Higher Multiplicity upon Restriction from GL(n) to GL(n−1)

Abstract: 

Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal series representation of $\GL_n(F)$ induced from a cuspidal representation of a Levi subgroup. When $\pi$ is an essentially square integrable representation of $\GL_{n-1}(F)$ we prove that $\Hom_{\GL_{n-1}}(\Pi,\pi)$  $= \mathbb{C}$ and $\Ext^i_{\GL_{n-1}}(\Pi,\pi) = 0$ for all integers $i\geq 1$, with exactly one exception (up to twists), namely, when $\Pi= \nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times \nu^{(\frac{n-1}{2})}$ and $\pi$ is the  Steinberg. When $\Pi= \nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times \nu^{(\frac{n-1}{2})}$ and $\pi$ is the Steinberg of $\GL_{n-1}(F)$, then $\dim \Hom_{\GL_{n-1}(F)}(\Pi,\pi)=n$. We also exhibit specific principal series for which each of the intermediate multiplicities $2, 3, \cdots, (n-1)$ are attained.

Along the way, we also give a complete list of those irreducible non-generic representations of $\GL_{n}(F)$ that have the Steinberg of $\GL_{n-1}(F)$ as a quotient upon restriction to $\GL_{n-1}(F)$. We also show that there do not exist non-generic irreducible representations of $\GL_{n}(F)$ that have the generalized Steinberg as a quotient upon restriction to $\GL_{n-1}(F)$.

Lecture series on Hodge Theory

Wednesday and Thursday

13 and 14 September, 11:30 am – 1.00 pm

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas
Affiliation: IIT Bombay

Title: Introduction to Hodge Theory

Abstract: These are part of an ongoing series of lectures on the basics of Hodge theory.

We will finish the proof of the de Rham theorem, via sheaf cohomology, and discuss some linear algebra needed for the Hodge theory, as in Chapter 1 of Huybrechts' book.

CACAAG seminar

Wednesday, 13 September, 5:30 PM

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Venue: Ramanujan Hall.

Host: Madhusudan Manjunath

Speaker: Madhusudan Manjunath, IIT Bombay

Title: The Chow Ring of a Simplicial Toric Variety IV.

Abstract: We will continue our study of the Chow ring of a simplicial toric variety.