Trimester II

From 1st January 2024  to 30th April 2024

Seminars in January 2022

Week 3 

3 PM (IST), 15th January 2024, Monday, at Zoom and YT Livestream.

Title: Linear Algebraic Groups over Algebraically Closed Fields.   (Slides are here and video is here)

Note: Neither of these two talks (especially, the first one) is meant for experts.
On the other hand, the audience is expected to have some familiarity with basic matrix groups.

Speaker: B. Sury, ISI Bangalore. 

Chair for the Talk: Gadadhar Misra, IIT Gandhinagar.

Abstract: This is a quick review of the theory of linear algebraic groups over an algebraically closed fields. The layout of the terrain will be explored with brief indications of reasoning for various results.

3 PM(IST), 18th January 2024, Thursday, at Zoom and YT Livestream.

Title: Tits's Classification theorem and the Tits Index.   (Slides are here and video is here)

Speaker: B. Sury, ISI Bangalore. 

Chair for the Talk:  Shanta Laishram, ISI Delhi.

Abstract: After recalling the Borel-Tits structure theory of linear algebraic groups over general fields, the Tits classification and the Tits index are described. Examples are discussed which show how one works with the Tits index.

Week 4

3 PM (IST), 23rd  January 2024, Tuesday, at Zoom and YT Livestream.

Title: Local Fields, Adele Rings and Idele Groups.  (Slides are here and video is here)

Speaker: Kartikeya Rai, University of Cambridge.

Chair for the Talk: Lalit Vaishya, IMSc. Chennai.

Abstract: We begin with a review of the basic structure theory for local fields à la André Weil. Next, we will describe restricted direct products, and use them to define and study adeles and ideles over a number field. These locally compact groups are particularly amenable to tools of harmonic analysis and are key players in much of modern number theory. Time permitting, we will also see idelic versions of the two major results of classical algebraic number theory: the unit theorem of Dirichlet, and the finiteness of the class number.

3 PM(IST), 25th  January 2024, Thursday, at Zoom and YT Livestream.

Title: Reductive Groups over General Fields.  (Slides are here and video is here)

Speaker: Mihir Sheth, IISc Bangalore.

Chair for the Talk: Sazzad Ali Biswas, SRM Univeristy AP.

Abstract: I will sketch the theory of reductive groups and their structural properties,
first over algebraically closed fields and then over general fields, focusing mainly on examples.

Week 5

3 PM(IST), 31st January 2024, Wednesday, at Zoom and YT Livestream.

Title: Harish-Chandra's Proof of Plancherel Theorem for SL(2, R).  (Slides are here and video is here)

Speaker: E. K. Narayanan, IISc Bangalore. 

Chair for the Talk: Ajay Kumar, Delhi University.

Abstract: Basics of SL(2, R), Infinite dimensional unitary representations, construction of principal series, discrete series and complementary series, infinitesimal method, classification of irreducible representations of SL(2, R), spherical functions,
spherical inversion and Plancherel theorem.

References:

1) V. Bargmann: Irreducible unitary representations of the Lorentz group, Ann. of Math.,

2 (48) 1947 568-640.

2) Serge Lang: SL(2, R).

3) A. W Knapp : Representation theory of semisimple Lie groups. An overview based on examples.

4) Harish-Chandra: Plancherel formula for the 2 x 2 real unimodular group. Proc. Nat. Acad.Sci., 38, 1952, 337-342.

Seminars in February.

Week 1

3 PM (IST), 1st February 2024, Thursday, at Zoom and YT Livestream.

Title: Harish-Chandra's Proof of Plancherel Theorem for SL(2, R).  (Slides are here and video is here)

Speaker: E. K. Narayanan, IISc Bangalore. 

Chair for the Talk: Gadadhar Misra, IIT Gandhinagar.

Abstract: Basics of SL(2, R), Infinite dimensional unitary representations, construction of principal series, discrete series and complementary series, infinitesimal method, classification of irreducible representations of SL(2, R), spherical functions,
spherical inversion and Plancherel theorem.

References:

1) V. Bargmann: Irreducible unitary representations of the Lorentz group, Ann. of Math.,

2 (48) 1947 568-640.

2) Serge Lang: SL(2, R).

3) A. W Knapp : Representation theory of semisimple Lie groups. An overview based on examples.

4) Harish-Chandra: Plancherel formula for the 2 x 2 real unimodular group. Proc. Nat. Acad.Sci., 38, 1952, 337-342.

3 PM (IST), 6th February 2024, Tuesday, at Zoom and YT Livestream.

Title: Harish-Chandra's Proof of Plancherel Theorem for SL(2, R).  (Slides are here and video is here)

Speaker:  E. K. Narayanan, IISc Bangalore. 

Chair for the Talk: Chandrasheel Bhagwat, IISER Pune.

Abstract: Same as the previous talk by speaker on 1st February. 

Week 2

3 PM (IST), 7th February 2024, Wednesday, at Zoom and YT Livestream.

Title: Harish-Chandra's Proof of Plancherel Theorem for SL(2, R). (Slides are here and video is here)

Speaker: E. K. Narayanan, IISc Bangalore. 

Chair for the Talk: V. Muruganandam, IIT Palakkad.

Abstract: Same as the previous talk by speaker on 1st February.

Week 3

3 PM (IST), 14th February 2024, Wednesday, at Zoom and YT Livestream.

Title: Harish-Chandra's Proof of Plancherel Theorem for SL(2, R).  (Slides are here and video is here)

Speaker: E. K. Narayanan, IISc Bangalore. 

 Chair for the Talk: Jotsaroop Kaur, IISER Mohali. 

 Abstract: Same as the previous talk by speaker on 1st February.

3 PM (IST), 16th February 2024, Friday, at Zoom and YT Livestream.

Title: Representations of p-adic Groups - I. (Slides are here and video is here)

Speaker: Mihir Sheth, IISc Bangalore.  

Chair for the Talk: Sazzad Ali Biswas, SRM University AP.

Abstract: The first talk in this series will introduce several basic notions in the smooth representation theory of locally profinite groups such as admissibility, Schur's lemma, characters, Hecke algebra, contragredient representation etc. We will try to provide some motivation and discuss a few examples.

Week 4

   3 PM (IST), 20th February 2024, Tuesday, at Zoom and YT Livestream.

Title: Representations of p-adic Groups - II. (Slides are here and video is here)

Speaker: Mihir Sheth, IISc Bangalore. 

Chair for the Talk: Rakesh Pawar, UMPA, ENS de Lyon.

Abstract: In the second talk, we will continue our discussion on smooth representations of p-adic groups by discussing parabolic induction, compact induction, Jacquet modules, supercuspidal representations, and the admissibility of irreducible representations.

7 PM (IST), 23rd February 2024, Friday, at Zoom and YT Livestream.

Title: Representation Theory and Automorphic Functions - I. (Slides are here and video is here)

Speaker: Surya Teja Gavva, CUNY New York.

Chair for the Talk: Harshvardhan Reddy, TIFR Mumbai.

Abstract: We will start with classical automorphic functions and modular forms and their connection to representation theory of GL(2, R) (and GL(2, A)). We will see how classical notions of holomorphic modular forms, Maass forms, relate to representation theoretic notions like irreducibility, Casimir, Raising/Lowering operators. If time permits, we will also discuss theta functions and their lifting to appropriate symplectic groups.

Week 5

3 PM, 27th February 2024, Tuesday at Zoom and YT Livestream.

Title: Discrete Series Representations of Real Semisimple Lie Groups  (video is here)

Speaker:  Chandrasheel Bhagwat, IISER Pune.

Chair for the Talk:  Anisa Chorwadwala, IISER Pune. 

Abstract: What is a discrete series representation of a real Lie group? When does it exist? (the condition rank G = rank K)

The story of discrete series for SL(2,R): Construction, properties (lowest weight vectors), 

Connection of discrete series of SL(2,R) with cusp forms.

3 PM, 28th February 2024, Wednesday, at Zoom and YT Livestream.

Title: Bernstein decomposition, Types and Hecke Algebras.  (video is here)

Speaker: Manish Mishra, IISER Pune.

Chair for the Talk: Amiya Kumar Mondal, IISER Berhampur.

Abstract: Let R(G) denote the category of smooth complex representation of G(F), where G is a connected reductive group defined over a non-archimedean local field F. Bernstein decomposition expresses R(G) as a product of indecomposable subcategories called Bernstein blocks. Each Bernstein block is equivalent to the module category of a canonically defined algebra (namely the endomorphism algebra of a "Bernstein projector"). It is also equivalent to the module category of the "Hecke algebra" associated with that "type".

In my first two talks, I will go over the basic theory mentioned above. I will then explain how these two algebra are related.  Moy-Prasad theory associates a number called depth to each Bernstein block. In the last talk, I will describe the depth-zero Hecke algebras and explain how these relate to the general case. 

Seminars in March.

Week 2

3 PM, 6th March 2024, Wednesday, at Zoom and YT Livestream.

Title: Bernstein decomposition, Types and Hecke Algebras.   (video is here)

Speaker:  Manish Mishra, IISER Pune.

Chair for the Talk: Mihir Sheth, IISc Bangalore. 

Abstract: Same as the previous talk by speaker on 28th February.

Week 4

3 PM, 20th March 2024, Wednesday, at Zoom and YT Livestream.

Title: Bernstein decomposition, Types and Hecke Algebras. (video is here)

Speaker: Manish Mishra, IISER Pune.

Chair for the Talk: Mihir Sheth, IISc Bangalore. 

Abstract: Same as the previous talk by speaker on 28th February.

3PM, 21st March 2024, Thursday, at Zoom and YT Livestream.

Title:  On Tate's thesis.      (Slides are here and video is here)

Speaker: Shiv Prakash Patel, IIT Dharwad.

Chair for the Talk: Surjeet Kour, IIT Delhi.

Abstract: Analytic continuation and functional equations of various L-functions are quite fascinating and it has been of central interest to number theorists. In his thesis, John Tate gave a more theoretical framework to derive analytic properties of L-functions of a number field using harmonic analysis on certain locally compact abelian groups attached to a number fields, namely Adels and Ideles. We review some basic properties of these groups. We sketch the proof of analytic continuation of zeta integrals following Tate's thesis.

7PM, 22nd March 2024, Friday, at Zoom and YT Livestream.

Title: Representation Theory and Automorphic Functions - II. (video is here)

Speaker: Surya Teja Gavva, CUNY New York.

Chair for the Talk: Tanusree Khandai, IISER Mohali. 

Abstract: We introduced classical notions of holomorphic and Maass forms in the last lecture. We will lift these forms to the group and connect them to the representation theory of GL_2. This will allow us to see all the special functions, differential equations, transforms etc more naturally, and allow us to use different coordinates/models that simplify computations. Further we lift to adeles to see how all the local places fit together. This will help in factorization of objects into local objects and view quantities at the infinite places like Laplace Eigenvalues, Gamma factors, etc and the local quantities like Hecke Eigenvalues, local factors etc at the same footing. 

Seminars in April.

Week 1

3PM, 2nd April 2024, Tuesday, at Zoom and YT Livestream.

Title: An introduction to Borel-Harish-Chandra theory. (Slides are here and video is here)

Speaker: Sandeep Varma, TIFR Mumbai.  

Chair for the Talk:  S. A. Katre, BP, Pune.

Abstract:  In this lecture series, we will first discuss the finiteness of volume as well as the non-compactness of $SL_m(\mathbb{Z}}) \backslash SL_m(\mathbb{R})$, using Minkowski reduction and Mahler's compactness theorem. We will then introduce congruence and arithmetic subgroups, the former also adelically. Then we will discuss some results on when $G(\mathbb{Q} \backslash G(\mathbb{A})$ is compact, where $G$ is an algebraic group over $\mathbb{Q}$ (or more generally, a number field), using adelic versions of Minkowski reduction and Mahler's compactness theorem. We then hope to discuss Siegel sets and approximate fundamental domains, and at least state results on and related to finiteness of volume for $G(\mathbb{Q} \backslash G(\mathbb{A})$. If time permits, we will discuss some applications.

3PM, 4th April 2024, Thursday, at Zoom and YT Livestream.

Title: An introduction to Borel-Harish-Chandra theory.  (Slides are here and video is here)

Speaker: Sandeep Varma, TIFR Mumbai. 

Chair for the Talk: Sazzad Ali Biswas, SRM University AP.

Abstract: Same as the previous talk by speaker on 2nd April.

Week 2

3PM, 9th April 2024, Tuesday, at Zoom and YT Livestream.

Title: An introduction to Borel-Harish-Chandra theory.  (Slides are here and video is here)

Speaker: Sandeep Varma, TIFR Mumbai. 

Chair for the Talk: Shanta Laishram, ISI Delhi.

Abstract: Same as the previous talk by speaker on 2nd April.

3PM, 11th April 2024, Thursday, at Zoom and YT Livestream.

Title: An introduction to Borel-Harish-Chandra theory.   (Slides are here and video is here)

Speaker: Sandeep Varma, TIFR Mumbai. 

Chair for the Talk: Somnath Jha, IIT Kanpur.

Abstract: Same as the previous talk by speaker on 2nd April.

Week 3

11 AM, 16th April 2024, Tuesday, at Zoom and YT Livestream.

Title: Bruhat-Tits Theory - I.       (Slides are here and video is here)

Speaker: Radhika Ganpathy, IISc Bangalore 

Chair for the Talk: Manish Mishra, IISER Pune.

Abstract: In this lecture, we will introduce Bruhat-Tits theory through two examples: the case of SL(2) and SU(3). We will introduce the standard apartment, affine root system, parahoric subgroups, the affine Weyl group,  and explain how the building is constructed in these two examples. We will also explain what the Moy-Prasad filtrations of parahoric subgroups are in these two examples.

11 AM, 18th April 2024, Thursday, at Zoom and YT Livestream.

Title: Bruhat-Tits Theory - II.        (Slides are here and video is here)

Speaker: Radhika Ganpathy, IISc Bangalore 

Chair for the Talk: Srilakshmi K, IISER Trivandrum.

Abstract: In this lecture, we will give an overview of the construction of the Bruhat-Tits building  of a split reductive p-adic group. The analogues of all the objects introduced in the first lecture via examples, will be discussed in the general setting. We will touch upon the construction of the building in the quasi-split case if time permits.

Week 4

3PM, 23rd April 2024, Tuesday, at Zoom and YT Livestream.

Title: The Finite-dimensionality of Spaces of Automorphic Forms - I.   (Slides are here and video is here)

Speaker: Ravi Raghunathan, IIT Bombay.

Chair for the Talk: Manish Mishra, IISER Pune.

Abstract: Harish-Chandra proved that the space of automorphic forms on a reductive group $G$ (with $G({\mathbb C)) connected) of a given $K$-type which are annihilated by an ideal $J$ of finite codimension in the centre of the universal enveloping algebra is finite dimensional. I will attempt to sketch a proof of this theorem.

3PM, 24th April 2024, Wednesday, at Zoom and YT Livestream.

Title: The Finite-dimensionality of Spaces of Automorphic Forms - II.    (Slides are here and video is here)

Speaker: Ravi Raghunathan, IIT Bombay.

Chair for the Talk: Shiv Prakash Patel, IIT Delhi.

Abstract: Same as the previous talk by speaker on 23rd April.