NCMW  Lie Groups and Representation Theory (2024)
Speakers and Syllabus
Syllabus to be covered in terms of modules of 6 lectures each
Name of the Speaker with their Affiliation 
No. of Lecture Hours 
Detailed Syllabus 
Dr. Santosh Nadimpalli, & Dr. Amiya Mondal IISER Berhampur 
6*1.5h = 9h 
Title: Induced representations and Intertwining operators. Topics:

Prof. C. S. Rajan, 
4*1.5h = 6h 
Title: Eisenstein series and Lfunction for rank one groups. Topics: Eisenstein series on GL2(R) for a lattice Γ and definition of constant term. Eisenstein series for rank one groups, constant term, mass Selberg relations, analytic continuation. Relation with Lfunctions. 
Prof. Pralay Chatterjee, 
3*1.5h=4.5h 
Title: Word maps on Lie groups and algebraic groups Topics: We will introduce the problems associated to the word maps in Lie groups with many examples. We will discuss the remarkable theorem of A. Borel on the Zariski density of images of word maps on semisimple algebraic groups, and the contrasting (relatively recent) results of A. Thom on compact Lie groups. We will also sketch some open problems in this area. 
Dr. Sandip Singh, 
3*1.5h=4.5h 
Title: Hypergeometric Groups and their Arithmeticity and Thinness Topics: A hypergeometric group Γ(f, g) is a subgroup of GLn(C) generated by the companion matrices of two monic coprime polynomials f and g of degree n. It arises as the monodromy group of a hypergeometric differential equation, and if the defining polynomials f and g are selfreciprocal and form a primitive pair, then the Zariski closure G of Γ(f, g) inside GLn(C) is either the symplectic group Spn or the orthogonal group On. It is immediate that when the defining polynomials f, g also have integer coefficients (in particular, when they are products of the cyclotomic polynomials), the associated hypergeometric group Γ(f, g) ⊆ G(Z). In this case, the hypergeometric group Γ(f, g) is called arithmetic if it has finite index inside G(Z), and thin, otherwise. The question of determining the arithmeticity and thinness of the hypergeometric groups has been of great interest since the beginning of the last decade. In this minicourse, we will provide all the details about the hypergeometric groups and discuss the progress in answering the arithmeticity and thinness question for hypergeometric groups. 
Dr. Pranab Sardar, 
4*1.5h=6h 
Title: Aspects of discrete subgroups of SO(n,1) and some questions Topics: We will discuss various propertiesofandquestions on discrete subgroups of SO(n,1) starting with the motivating examples from the cases n = 2. 
Prof. Krishnendu Gongopadhyay, 
3*1.5h=4.5h 
Title:Reversibility in Lie groups Topics: Reversible or real elements in a group are those elements that are conjugate to their own inverses. Such elements appear naturally in different branches of mathematics. They are closely relatedtostronglyreversible or strongly real elements, which can be expressed as a product of two involutions. Classifying reversible and strongly reversible elements in groups has been a problem of broad interest.We will discuss some recent developments on this topic. 
In addition to the above main courses, we shall also have the following research expository lectures to expose the participants to some of the current developments invarious directions on the subject. 

Prof. Anish Ghosh, TIFR, Mumbai 
1.5h 
Title: Random walks: dynamics and geometry Abstract: I will discuss random walks on homogeneous spaces of Lie groups and explain connections with geometry and dynamics, especially on flat surfaces. 
Dr. Arunava Mandal, IIT Roorkee 
1.5h 
Title: On some stable subgroups under Lie group automorphisms Topics: In a seminal work in 1955 by A. Borel and G. D. Mostow on semisimple automorphisms of Lie algebras facilitates proving various results on certain stable subgroups of a Lie algebra under the group of automorphisms. It has a significant impact on a wide range of mathematical studies, such as the invariance of maximal tori, Cartan subgroups, Borel subgroups, which are always subgroups of interest to a large extent in the theory of algebraic groups and Lie groups. Here we will give a survey on this topic. 
Dr. Arghya Mondal, Krea University 
1.5h 
Title: Applications of Property (T) and related notions
Topics: Property (T) of a group has found many applications. We will discuss why and where it is useful. 
Dr. Tanusree Khandai, IISER Mohali 
1.5h 
Title: Highest Weight Representations of Semisimple Lie algebras Topics: The Lie algebra $\mathfrak{gl}_n(\mathbb C)$ can be decomposed in an obvious way into sum of three subalgebras : the strictly upper triangular matrices, strictly lower triangular matrices and the diagonal matrices. Several classes of Lie algebras including the split semisimple Lie algebras, KacMoody Lie algebras and Virasoro Lie algebras admit such a decomposition. With respect to this decomposition one can talk about the highest weight representations for Lie algebras with triangular decomposition. The Verma modules are the universal objects in the category of highest weight representations. With $\mathfrak{sl}_2(\mathbb C)$ as our primary example of a Lie algebra with triangular decomposition, we shall introduce the notion of Verma modules and discuss its properties. 
References:
 J. P. Serre, Linear representations of finite groups.
 Anthony W. Knapp, Representation theory of semisimple groups.
 Automorphic forms, representations and Lfunctions, Volume1 Editors Armand Borel and W. Casselman, (Corvallis, Oregon).
 Armand Borel, Automorphic forms on SL(2, R).
 Freydoon Shahidi, Eisenstein series and automorphic Lfunctions.
 Borel, A. On free subgroups of semisimple groups. Enseign. Math. (2) 29 (1983), no. 12, 151–164.
 B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s property (T), New Math. Monogr.,11, Cambridge University Press, Cambridge, 2008. xiv+472 pp.
 D. Y. Kleinbock, G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999) 451–494.
 A. Borel, G.D. Mostow, On semisimple automorphisms of Lie algebras. Ann. of Math. (2) 61 (1955), 389–405, MR0068531.
 A. Borel and G. D. Mostow,: On semisimple automorphisms of Lie algebras, Ann. of Math. 61 (1955), 389504.
 K Gongopadhyay, C Maity, Reality of Unipotent Elements in Classical Lie Groups, Bulletin des Sciences Mathématiques, 185, July 2023, 103261.
 K. Gongopadhyay, T. Lohan, C. Maity, Reversibility of affine transformations, Proceedings of the Edinburgh Mathematical Society, 2023, 66, 1217–1228,
 A. G. O’Farrell, I. Short, Reversibility in Dynamics and Group Theory, London Mathematical Society Lecture Note Series, vol.416, Cambridge University Press, Cambridge, 2015
 J. Bajpai, S. Singh, On orthogonal hypergeometric groups of degree five, Trans. Am. Math. Soc. 372 (11) (2019) 7541–7572. MR4029673
 F. Beukers, G. Heckman, Monodromy for the hypergeometric function nFn−1 Invent. Math. 95 (2) (1989) 325–354. MR0974906
 C. Brav, H. Thomas, Thin monodromy in Sp(4), Compos. Math. 150 (3) (2014) 333–343. MR3187621
 E. Fuchs, C. Meiri, P. Sarnak, Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, J. Eur. Math. Soc. (JEMS) 16 (8) (2014) 1617–1671. MR3262453
 P. Sarnak, Notes on Thin Matrix Groups. Thin Groups and Superstrong Approximation, Math. Sci. Res. Inst. Publ., vol. 61, Cambridge Univ. Press, Cambridge, 2014, pp. 343–362. MR3220897
 S. Singh, Arithmeticity of four hypergeometric monodromy groups associated to CalabiYau threefolds, Int. Math. Res. Not. (IMRN) 2015 (18) (2015) 8874–8889. MR3417696
 S. Singh, T.N. Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163 (3) (2014) 591–617. MR3165424
Time Table
Day 
Date 
Lecture 1 (9.30 
Tea 
Lecture 2 (11.30 
Lunch 
Lecture 3 (2.00–3.30) 
Tea 
Tutorial/Lecture 
Snacks 


(name of the speaker) 

(name of the speaker) 

(name of the speaker) 

(name of the speaker) 

Mon 
16/12 
SN 
Tea & snacks 
PC 
Lunch Break 
AM 
Tea 
KG 
Tea & snacks 
Tue 
17/12 
SN 
Tea & snacks 
PC 
Lunch Break 
AMo 
Tea 
Tutorial 
Tea & snacks 
Wed 
18/12 
SN 
Tea & snacks 
PC 
Lunch Break 
KG 
Tea 
AM 
Tea & snacks 
Thu 
19/12 
SN 
Tea & snacks 
SS 
Lunch Break 
KG 
Tea 
Tutorial 
Tea & snacks 
Fri 
20/12 
CSR 
Tea & snacks 
SS 
Lunch Break 
SD 
Tea 
AMa 
Tea & snacks 
Sat 
21/12 
CSR 
Tea 
SS 
Lunch Break 
SD 
Tea 
Tutorial 
Tea 
Sun 
22/12 
FREE DAY 

Mon 
23/12 
CSR 
Tea 
AG 
Lunch Break 
PS 
Tea 
Tutorial 
Tea 
Tue 
24/12 
CSR 
Tea 
TK 
Lunch Break 
PS 
Valedictory 
Full forms for the abbreviations of speakers and tutors:
PC: Pralay Chatterjee
SD: Subhabrata Das AG: Anish Ghosh
KG: Krishnendu Gongopadhyay TK: Tanusree Khandai
AMa: Arunava Mandal
AM: Amiya Mondal
AMo: Arghya Mondal
SN: Santosh Nadimpalli
CSR: C. S. Rajan
PS: Pranab Sardar
SS: Sandip Singh
Possible tutors :
1. Arindam Jana, IISER Berhampur
2. Chandan Maity, IISER Berhampur
3. Tejbir Lohan, IIT Kanpur
4. Arghya Mondal, Krea University
5. Shashank Vikram Singh, IISER Mohali
6. Rakesh Halder, IISER Mohali