


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 nonarchimedean 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 $\GL_{n1}(F)$ we prove that $\Hom_{\GL_{n1}}(\Pi,\pi)$ $= \mathbb{C}$ and $\Ext^i_{\GL_{n1}}(\Pi,\pi) = 0$ for all integers $i\geq 1$, with exactly one exception (up to twists), namely, when $\Pi= \nu^{(\frac{n1}{2})} \times \nu^{(\frac{n3}{2})} \times \ldots \times \nu^{(\frac{n1}{2})}$ and $\pi$ is the Steinberg. When $\Pi= \nu^{(\frac{n1}{2})} \times \nu^{(\frac{n3}{2})} \times \ldots \times \nu^{(\frac{n1}{2})}$ and $\pi$ is the Steinberg of $\GL_{n1}(F)$, then $\dim \Hom_{\GL_{n1}(F)}(\Pi,\pi)=n$. We also exhibit specific principal series for which each of the intermediate multiplicities $2, 3, \cdots, (n1)$ are attained.
Along the way, we also give a complete list of those irreducible nongeneric representations of $\GL_{n}(F)$ that have the Steinberg of $\GL_{n1}(F)$ as a quotient upon restriction to $\GL_{n1}(F)$. We also show that there do not exist nongeneric irreducible representations of $\GL_{n}(F)$ that have the generalized Steinberg as a quotient upon restriction to $\GL_{n1}(F)$.
Lecture series on Hodge Theory
Wednesday and Thursday
13 and 14 September, 11:30 am – 1.00 pm

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.