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,
IIT Kanpur

&

Dr. Amiya Mondal IISER Berhampur

6*1.5h = 9h

Title: Induced representations and Intertwining operators.

Topics:

  • Formalism of induced representations, intertwining operators in the setting of finite groups with finite groups of low rank as principal example.

  • Induced representations in the context of real groups such as SL2(R), SL2(C) and SO(n,1). We will introduce the setting in the p-adic case as well.

  • Intertwining operators of the above induced representations and their analytic continuation

  • Properties of intertwining operators such as, admissibility, unitarity, poles, reducibility points, and complementary series.

Prof. C. S. Rajan,
Ashoka University

4*1.5h = 6h

Title: Eisenstein series and L-function 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 L-functions.

Prof. Pralay Chatterjee,
IMSc Chennai

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.

Prof. T. N.Venkataramana,
ICTS, Bengaluru

3*1.5h=4.5h

Title: Arithmeticity of higher rank lattices.

Topics: In 1984, a translation into English of a paper of Margulis on arithmeticity of lattices in higher rank groups was published in the Inventiones. The proof there is quite accessible and self-contained. It uses measure theory in an essential way. We review the proof developing the necessary background.

 

Dr. Sandip Singh,
IIT Bombay

 

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 self-reciprocal 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 mini-course, 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,
IISER Mohali

 

2*1.5h=3h

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.

Dr. Subhabrata Das, Presidency University

2*1.5h = 3h

Title: Property Rapid Decay for Discrete Groups

Topics:

  • Equivalent formulations of Property Rapid Decay (P(RD)).
  • Brief outline of the theorems: (a) Free groups have P(RD) (Haagerup), (b) co-compact lattices in SL(3,R) and SL(3,C) have P(RD) (Lafforgue).
  • A short survey of results around Valette's conjecture: co-compact lattices in semisimple Lie groups have P(RD).

 

Prof. Krishnendu Gongopadhyay,
IISER Mohali

 

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, Kac-Moody 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:

  1. J. P. Serre, Linear representations of finite groups.
  2. Anthony W. Knapp, Representation theory of semisimple groups.
  3. Automorphic forms, representations and L-functions, Volume-1 Editors- Armand Borel and W. Casselman, (Corvallis, Oregon).
  4. Armand Borel, Automorphic forms on SL(2, R).
  5. Freydoon Shahidi, Eisenstein series and automorphic L-functions.
  6. Borel, A. On free subgroups of semisimple groups. Enseign. Math. (2) 29 (1983), no. 1-2, 151–164.
  7. B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s property (T), New Math. Monogr.,11, Cambridge University Press, Cambridge, 2008. xiv+472 pp.
  8. D. Y. Kleinbock, G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999) 451–494.
  9. A. Borel, G.D. Mostow, On semi-simple automorphisms of Lie algebras. Ann. of Math. (2) 61 (1955), 389–405, MR0068531.
  10. A. Borel and G. D. Mostow,: On semi-simple automorphisms of Lie algebras, Ann. of Math. 61 (1955), 389-504.
  11. K Gongopadhyay, C Maity, Reality of Unipotent Elements in Classical Lie Groups, Bulletin des Sciences Mathématiques, 185, July 2023, 103261.
  12. K. Gongopadhyay, T. Lohan, C. Maity, Reversibility of affine transformations, Proceedings of the Edinburgh Mathematical Society, 2023, 66, 1217–1228,
  13. 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
  14. J. Bajpai, S. Singh, On orthogonal hypergeometric groups of degree five, Trans. Am. Math. Soc. 372 (11) (2019) 7541–7572. MR4029673
  15. F. Beukers, G. Heckman, Monodromy for the hypergeometric function nFn−1 Invent. Math. 95 (2) (1989) 325–354. MR0974906
  16. C. Brav, H. Thomas, Thin monodromy in Sp(4), Compos. Math. 150 (3) (2014) 333–343. MR3187621
  17. 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
  18. 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
  19. S. Singh, Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau three-folds, Int. Math. Res. Not. (IMRN) 2015 (18) (2015) 8874–8889. MR3417696
  20. 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
(9.30–11.00) 		Tea
(11.05–11.25) 		Lecture 2
(11.30–1.00) 		Lunch
(1.05–1.5 5) 		Lecture 3
(2.00–3.30) 		Tea
(3.35-3.55) 		Tutorial \Lecture
(4.00-5.30) 		Snacks
5.30-6.00
    (name of the speaker)   (name of the speaker)   (name of the speaker)   (Name of the tutor)  
Mon 16/12 TNV Tea & snacks PC Lunch Break SN Tea AmM Tea & snacks
Tue 17/12 TNV Tea & snacks PC Lunch Break SN Tea Tutorial Tea & snacks
Wed 18/12 TNV Tea & snacks PC Lunch Break AmM Tea KG Tea & snacks
Thu 19/12 SN Tea & snacks CSR Lunch Break AruM Tea Tutorial Tea & snacks
Fri 20/12 CSR Tea & snacks SS Lunch Break PS Tea SN Tea & snacks
Sat 21/12 CSR Tea & snacks SS Lunch Break PS Tea Tutorial Tea & snacks
Sun 22/12 F R E E D A Y
Mon 23/12 CSR Tea & snacks ArgM Lunch Break SD Tea Tutorial Tea & snacks
Tue 24/12 AG Tea & snacks TK Lunch Break SD Valedictory Tea & Snacks

 

 

 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
TVN: T. N. Venkataramana

Possible tutors :
    1. Arindam Jana, IISER Berhampur
    2. Chandan Maity, IISER Berhampur
    3. Tejbir Lohan, IIT Kanpur
    4. Abhishek Mukherjee, Kalna College
    5. Shashank Vikram Singh, IISER Mohali
    6. Rakesh Halder, IISER Mohali

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