Present Position

Senior Professor, School of Mathematics,

Tata Institute of Fundamental Research,

Colaba, Mumbai 400005, INDIA

PhD from Harvard University, 1989.

Mathematical Interest

• Algebraic number theory,
• Automorphic forms,
• Representation theory.

[List of Publications] [Description of Work] [Professional Recognition] [Professional Experience]

Editorial work

Managing Editor: International Journal of Number Theory 2005-2010.

Editor: Journal of Number Theory.

Editor: Math. Zeitschrift.

Editor: Journal of Ramanujan Mathematical Society.

Some recent invitations to lecture

1. Conference on Number Theory and Representation theory' at Harvard University on Dick Gross's 60th birthday, in June 2010.

2. Conference in Berlin on Wilhelm Zink's 65th birthday in July 2010.

3. Conference at RIMS, Kyoto, Sept 2010.

4. NISER Foundation day conference in Bhuvaneswar in Dec. 2010 Doing mathematics by asking questions: examples from Number Theory'.

5. Invited to speak on the occasion of SASTRA award to Wei Zhang in Kumbakonam in Dec. 2010.

6. Platinum Jubilee Award Lecture in the Indian Science Congress in Chennai in Jan 2011.

7. Lectured in IIT Kanpur on the Fundamental Lemma', the work of the Fields Medallist, B-C. Ngo in April 2011.

8. Special activity on Automorphic representations at Morning Side Center in Beijing in May-June 2011.

9. An FRG meeting at University of Colarado in June 2011.

10. Conference on $L$-packets in Banff on Relative Local Langlands conjecture', Canada in June 2011.

11. Conference in Max-Planck Institute, Germany in August 2011, on Representations of Lie groups'.

12. Lectures in the workshop on Deligne-Lusztig theory at TIFR, Mumbai in December 2011.

13. Conference at National University of Singapore, on branching laws, in particular on Gross-Prasad' conjectures in March 2012.

14. Symposium at Panjab University, Chandigarh; lectured on Modelling representation theory' on Feb 7, 2012.

15. Madan Mohan Malaviya's One hundred fiftieth annivarsay lecture on Groups as Unifying themes' at BHU, Varanasi on Feb. 11, 2012

16. Workshop Representations des groupes reductifs p-adiques' in Porquerolles island, near Toulon, FRANCE from 17 to 23 June 2012. Lectured on Ext-analogue of branching laws for Classical groups'.

17. ARCC workshop "Hypergeometric motives" at ICTP, Trieste from June 21st to June 30th. Lectured on Automorphic representations, Motives, and L-functions'.

18. Gave INSPIRE lectures at Shivaji University, Kolhapur in May 2012, on A perspective on Mathematics through examples'.

19. Gave lectures in the AIS program at IISER, Mohali on Representation theory of finite groups of Lie type: Deligne-Lusztig theory'.

20. Plenary speaker at Ramanujan Mathematical Society meeting in Delhi in Oct. 2012, on Fourier coefficients of Automorphic forms'.

21. Plenary speaker at the Legacy of Ramanujan conference in Delhi in Dec. 2012 on Fourier coefficients of Automorphic forms'.

22. Gave INSPIRE lecture at Kumaun University, Nainital in Dec. 2012 on A perspective on Mathematics through examples'.

23. Gave INSPIRE lecture at Guru Nanak Khalsa College, Matunga in Oct 2012, on An overview of mathematics through examples'.

24. Gave a series of lectures in Pune University on Ramanujan Graphs and Number theory' in March, 2013.

25. Oberwolfach Workshop: Spherical Varieties and Automorphic Representations, May 12th to 18th, 2013.

26. Tsinghua University, Beijing in June 2013.

27. Lectured in the DST-JSPS conference in Tokyo in November 2013 on Branching laws and the local Langlands correspondence'.

28. Gave a colloquium lecture in Tokyo University in November 2013 on Ext Analogues of Branching laws'.

29. Gave an invited talk in the conference at IMSc on occasion of Ram Murty's 60th birthday in December 2013 on  Counting integral points in a polytope: a problem in invariant theory'.

30. Lectured in the Inaugural Conference in Sanya, China in December 2013 On distinguished representations'.

31. Lectured in an Advanced Instructional School on IISER, Pune in December 2013 on Maximal subgroups of classical groups'.

32. Lectured in the Conference in Oberwolfauch in January 2014 on Ext-Analogues of branching laws'.

33. Lectured in a workshop at HRI, Allahabad in March 2014 on Schur Multiplier for finite, real and p-adic groups'.

34. Invitation to the conference in Banff June 01-07, 2014 on Future of trace formula'.

35. Summer school in Jussieu in June 2014 on Gan-Gross-Prasad conjectures'.

36. Conference at Univ. Paris 13 in June 2014.

37. Research Professor at MSRI August to Dec. 2014.

Papers in Journals

 [PDF file] (With B.H. Gross) Test Vectors for linear forms, Maths Annalen, vol. 291, 343-355 (1991). PDF file. Invariant linear forms for representations of GL(2) over a local field, American J. of Maths, vol. 114, 1317-1363 (1992). [PDF file] (With B.H. Gross) On the decomposition of a representation of SO(n) when restricted to SO(n-1), Canadian J. of Maths, vol. 44, 974-1002 (1992). [PDF file] On the decomposition of a representation of GL(3) restricted to GL(2), Duke J. of Maths, vol. 69, 167-177 (1993). PDF file. Bezout's theorem for simple abelian varieties, Expositiones Math, vol. 11, 465-467 (1993). PDF file. On the local Howe duality correspondence, IMRN, No. 11, 279-287 (1993). [PDF file] (With B.H. Gross) On irreducible representations of SO(2n+1)xSO(2m), Canadian J. of Maths, vol. 46(5), 930-950 (1994). [PDF file] On an extension of a theorem of Tunnell, Compositio Math., vol. 94, 19-28(1994). PDF file. (With D. Ramakrishnan) Lifting orthogonal representations to spin groups and local root numbers, Proc. of the Indian Acad. of Science, vol. 105, 259-267 (1995). [PDF file] Some applications of seesaw duality to branching laws, Maths Annalen, vol. 304, 1-20 (1996). [PDF file] (With C. Khare) Extending local representations to global representations, Kyoto J. of Maths, vol. 36, 471-480 (1996). [PDF file] On the self-dual representations of finite groups of Lie type, J. of Algebra, vol. 210, 298-310 (1998). [PDF file] Some remarks on representations of a division algebra and of Galois groups of local fields, J. of Number Theory, vol. 74, 73-97 (1999). [PDF file] Distinguished representations for quadratic extensions, Compositio Math., vol. 119(3), 343-354 (1999). [PDF file] (with D. Ramakrishnan) On the global root numbers of $GL(n) \times GL(m)$, Proceedings of Symposia in Pure Maths of the AMS, vol. 66, 311-330, (1999). [PDF file] On the self-dual representations of $p$-adic groups, IMRN vol. 8, 443-452 (1999). [PDF file] (With Kumar Murty) Tate cycles on a product of two Hilbert Modular Surfaces, J. of Number Theory, vol. 80, 25-43 (2000). [PDF file] Theta correspondence for Unitary groups, Pacific J. of Maths, vol. 194, no. 2, 427-438 (2000). [PDF file] (with A. Raghuram) Kirillov theory of $GL_2(D)$ where $D$ is a division algebra over a non-Archimedean local field, Duke J. of Math, vol. 104, no. 1, 19-44 (2000). [PDF file] Comparison of germ expansion for inner forms of $GL_n$, Manuscripta Mathematicae, vol. 102, 263-268 (2000). [PDF file] The space of degenerate Whittaker models for general linear groups over finite fields, IMRN, vol. 11, 579-595 (2000). [PDF file] (With C. Khare) On the Steinitz module and capitulation of ideals, Nagoya Math. J.,vol. 160, 1-15 (2000). [PDF file] (with C.S.Yogananda) Bounding the torsion in CM elliptic curves, Comptes Rendus Mathematiques Mathematical Reports of the Academy of Sciences, Canada, vol. 23, 1-5 (2001). [PDF file] On a conjecture of Jacquet about distinguished representations of $GL_n$, Duke J. of Math, vol. 109, 67-78 (2001). [PDF file] Locally algebraic representations of p-adic groups, appendix to the paper by P.Schneider and J.Teitelbaum, $U({\frak g})$-finite locally analytic representations, (Electronic Journal) Representation Theory, vol. 5, 111-128 (2001). [PDF file] (with Nilabh Sanat)} On the restriction of cuspidal representations to unipotent elements, Math. Proceedings of Cambridge Phil. Society, vol. 132(1), 35-56 (2002). [PDF file] (with CS Rajan)} On an Archimedean analogue of Tate's conjecture, J. of Number Theory, vol. 99 (2003), 180-184. [PDF file] (with UK Anandavardhanan) Distinguished representations for SL(2), Math. Res. Letters 10, 867--878 (2003). [PDF file] On an analogue of a conjecture of Mazur: A question in Diophantine approximation, Contributions to automorphic forms, geometry, and number theory, 699--709, Johns Hopkins Univ. Press, Baltimore, MD, 2004. [PDF file] (with C. Khare) Reduction of homomorphisms mod p, and algebraicity, J. of Number Theory, vol. 105, 322--332 (2004). [PDF file] (with SO Juriaans and IBS Passi) Hyperbolic Unit Groups, Proc. of the AMS, vol. 133, (2005) no. 2, 415-423. [PDF file] (with Jeffrey D. Adler) On certain mulitplicity one theorems, Israel J. of Mathematics, vol. 153, 221-245 (2006). [PDF file] (with UK Anandavardhanan) On the SL(2) period integral, American J. of Mathematics, vol. 128, 1429-1453 (2006). [PDF file] Relating invariant linear form and local epsilon factors via global methods, with an appendix by H. Saito; Duke J. of Math. vol. 138, No.2, 233-261 (2007). [PDF file] (with Rainer Schulze-Pillot) Generalised form of a conjecture of Jacquet, and a local consequence; Crelle Journal 616, 219-236 (2008). [PDF file] (with Dinakar Ramakrishnan) On the self-dual representations of division algebras over local fields; American J. of Math., vol. 134, no. 3, 749-772 (2012). [PDF file] (with Shrawan Kumar and George Lusztig) Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups. Contemporary Math., vol 478, AMS, pp. 99-101. [PDF file ] (with Ramin Takloo-Bighash) Bessel models for GSp(4); Crelle Journal, vol. 655, 189-243 (2011). [PDF file ] Some remarks on representations of quaternion division algebras. [PDF file] (with Wee Teck Gan and Benedict H. Gross) Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups; vol. 346, pp 1-109, Asterisque (2012) . [PDF file] (with Wee Teck Gan and Benedict H. Gross) Restriction of representations of classical groups: Examples; vol. 346, pp. 111-170, Asterisque (2012). [PDF file] (with U.K. Anandavardhanan) A local-global question in Automorphic forms; Compositio Math 149 (2013), no. 6, 959--995. [PDF file] (with Jeff Adler) Extensions of representations of $p$-adic groups, special volume of Nagoya J. of Math dedicated to the memory of Prof. Hiroshi Saito, vol. 208, pp. 171-199 (2012). [PDF file] (with Dinakar Ramakrishnan) On the cuspidality criterion for the Asai transfer to ${\rm GL}(4)$; an appendix to Determination of cusp forms on $GL(2)$ by coefficients restricted to quadratic subfields by M. Krishnamurthy;'' Journal of Number Theory , Volume 132, Issue 6, Pages 1359-1384 (June 2012) [PDF file] A relative' local Langlands conjecture. [PDF file] (with B. Gross and W.T. Gan) Branching laws: The non-tempered case. Ext versions of branching laws. [PDF file] (with Shrawan Kumar) Dimension of zero weight space: an algebro-geometric approach; Journal of Algebra , volume 403 (2014) 324-344. [PDF file] A character relationship on ${\rm GL}_n$; to appear in Israel Journal. [PDF file] Half the sum of positive roots, the Coxeter element, and a theorem of Kostant; to appear in Forum Math. [PDF file] (with Shiv Prakash Patel) Multiplicity formula for restriction of representations of $\widetilde{\GL_{2}}(E)$ to $\widetilde{\SL_{2}}(E)$; submitted. [PDF file]

Papers in Conference Proceedings

 [PDF file] Ribet's Theorem: Shimura-Taniyama-Weil implies Fermat, Proceedings of the seminar on Fermat's Last Theorem at Fields Institute, edited by V. Kumar Murty, CMS Conference Proceedings, vol. 17, 155-177 (1995). [PDF file] A brief survey on the Theta correspondence, Proceedings of the Trichy Conference edited by K. Murty and M. Waldschmidt, Contemporary Maths, AMS, vol. 210, 171-193 (1997). [PDF file] (with C.S.Yogananda) A report on Artin's holomorphy conjecture, in the volume on Number Theory, edited by R.P.Bambah, V.C.Dumir, and R.J.Hans-Gill, Hindustan Book Agency, (1999) 301-314. [PDF file] The space of degenerate Whittaker Models for $GL_4$ over $p$-adic fields, Proceedings of the TIFR conference on Automorphic Forms, AMS (2001). [PDF file] The main theorem of Complex Multiplication, Proceedings of the Advanced Instructional School on Algebraic Number Theory, entitled, Elliptic Curves, Modular Forms, and Cryptography", HRI, Allahabad (2000). [PDF file]

Unpublished Papers

 [PDF file ] Contributions to Algebraic number theory from India since Independence, unpublished. [PDF file]

Unpublished Lecture Notes

 [PDF file] Lectures on Algebraic Groups (2002), notes by Shripad Garge. [PDF file] (with A. Raghuram) Representation theory of $GL(n)$ over non-Archimedean local fields, lecture notes for a workshop at ICTP, Italy (2001). [PDF file] Lectures on Tate's thesis, lecture notes for a workshop at ICTP, Italy (2007). [PDF file] Some questions on representations of Algebraic Groups (2011). [PDF file] Notes on representations of finite groups of Lie type (2014). [PDF file] Notes on modular representations of $p$-adic groups, and the Langlands correspondence (2014). [PDF file]

A Brief Description of Work so far

Branching Laws for representations of Real and p-adic groups: Many problems in representation theory involve understanding how a representation of a group decomposes when restricted to a subgroup. Situations which involve multiplicity one phenomenon in which either the trivial representation, or some other representation of the subgroup appears with multiplicity at most one is specially useful. To cite a few examples, the theory of spherical functions and Whittaker models depends on such a multiplicity one phenomenon. The Clebsch-Gordon theorem about tensor product of representations of SU(2) has been very useful both in Physics and Mathematics. Many of my initial papers have been about finding such multiplicity one situations for infinite dimensional representations of real and $p$-adic groups. The results are expressed in terms of the arithmetic information which goes in parameterising representations, the so called Langlands parameters. In particular, the Clebsch-Gordon theorem was generalised by me for infinite dimensional representations of real and $p$-adic GL(2). Several papers, some written in collaboration with B.H.Gross, point out to the importance of the so called epsilon factors in these branching laws. The papers [1], [2], [3], [4], [5], [9], [11], [14] belong to this theme. These works have implication for the global theory of automorphic forms. There are many parallels between global period integrals, expressed in many situations as special value of $L$-functions, and local branching laws expressed in terms of epsilon factors. In paper [35] this theme has been carried out, giving a global proof of the decomposition of tensor product of two representations of GL(2) in terms of epsilon factors. In paper [36] written with Schulze-Pillot, we generalise Jacquet's conjecture to general cubic algebras, and deduce the local analogue. This paper also proves a very general globalisation theorem of local representations.
The paper [15] studies the question of when a representation of $G(K)$ has a $G(k)$-invariant vector for $K$ a quadratic extension of $k$ for $k$ either a finite or a $p$-adic field. In the $p$-adic case, this was done only for division algebras in [15]. I have used the methods of this paper to prove a conjecture of Jacquet about distinguished representations of $GL_n$ and $U_n$ in the case when $K$ is a unramified quadratic extension of $k$ in [25].
Lusztig followed up the theme of [15] in his paper in Representation Theory , vol.4, (2000).
I have written a paper [33] with Jeff. Adler in which we prove several multiplicity 1 theorems; in particular we show that an irreducible representation of $GSp(2n)$ when restricted to $Sp(2n)$ decomposes with mutiplicity 1 for $p$-adic fields.

Representations of division algebras and of Galois groups of local fields: Generalising local class field theory, Langlands has conjectured a correspondence between irreducible representations of $GL(n)$ or of a division algebra of index $n$ to $n$ dimensional representations of the Galois group of the local field. This correspondence has recently been established by Harris, Taylor and Henniart. The correspondence preserves self-dual representations. Self-dual representations are of two kinds: symplectic and orthogonal. The question is: how does the Langlands correspondence behave on these two kinds of self-dual representations. Based on considerations of Poincare duality on the middle dimensional cohomology of a certain rigid analytic space, Dinakar Ramakrishnan and I conjecture that a representation of division algebra is orthogonal if and only if the associated representation of the Galois group is symplectic. The conjecture was made in [10]. The paper [14] was also motivated by its consideration. In the paper [37] with Ramakrishnan we show how this conjecture is a consequence of `functoriality', and since the functorial lift between classical groups and $GL(n)$ is now known in many cases, we are able to prove the conjecture in [37] for those cases when the parameter is symplectic.

Self-dual representations of finite and $p$-adic groups : For a compact connected Lie group it is a theorem due to Malcev that an irreducible, self-dual representation carries an invariant symmetic or skew-symmetric bilinear form depending on the action of a certain element in the center of the group. We have generalised this result to finite groups of Lie type in [13] and to $p$-adic groups in [17], providing an answer to a question raised by Serre. These results are, however, proved only for generic representations and a condition on the group: the group contains an element which operates by $-1$ on all simple roots. The group $SL(n)$ for $n \cong 2 \bmod 4$ does not have such an element for a finite field ${\Bbb F}_q$ for $q \cong 3 \bmod 4$, and for such group there are generic self-dual representations on which the central element acts trivially, although the representation is symplectic, belying a belief at that point. A. Turull later gave much more complete results about Schur index in general for $SL(n)$.

Kirillov/Whittaker models : In the work [20] done with A. Raghuram, we develop Kirillov theory for irreducible admissible representations of $GL_2(D)$ where $D$ is a division algebra over a non-Archimedean local field. This work is in close analogy with the work of Jacquet-Langlands done in the case when $D$ is a field, and realises any irreducible admissible representation of $GL_2(D)$ on a space of functions of $D^*$ with values in what may be called the space of degenerate Whittaker models which is the largest quotient of the representation on which the unipotent radical of the minimal parabolic which is isomorphic to $D$ acts via a non-trivial character of $D$. Paper [22] studies this space of degenerate Whittaker models for finite fields obtaining a rather pretty result about the space of degenerate Whittaker model for a cuspidal representation of $GL_{2n}({\Bbb F})$ with respect to the $(n,n)$ parabolic with unipotent radical $M_n({\Bbb F})$. In paper [40] in the conference proceedings of a conference at the Tata Institute on Automorphic forms, I elaborate on a conjecture with B. Gross which gives a very precise structure for the space of degenerate Whittaker models on $GL_2(D)$ when $D$ is a quaternion division algebra. There is also a proposal in this paper to interpret triple product epsilon factors (for $GL(2)$) in terms of intertwining operators.

Weil Representations: Generalising the classical construction of theta functions, Weil representations provide one of the few general methods of constructing representations of groups over real and $p$-adic groups, as well as automorphic forms. The relation of this construction of representations to the Langlands parametrisation is still not fully understood. I have written two papers dealing with this question in which I refine some conjectures of Jeff Adams on the Langlands parameters of representations obtained via the Weil construction, thus making rather precise conjectures about the behaviour of the theta correspondence for groups of similar size. I have also done some work on the $K$-type of the Weil representation, and also on the character formula for the Weil representation. Papers [7], [19] as well as the expository paper [40] containing some new results too, belong to this theme.

Modular forms: There is a well known theorem of Deligne about estimates on the Fourier coefficients of modular forms. In the paper [12] with C. Khare, we study whether the converse is true, i.e. if given finitely many algebraic integers satisfying Deligne bounds, there exists an eigenform of Hecke operators with these algebraic integers as Fourier coefficients. One simple case of this problem is solved by an application of Wiles's theorem about the Shimura-Taniyama conjecture.

Representations of finite groups of Lie type: I have worked on some aspects of representation theory of finite groups of Lie type with my student Nilabh Sanat, and we have written a paper [27] together. This paper decomposes an irreducible cuspidal representation of a classical group restricted to its maximal unipotent subgroup as an alternating sum of certain explicit unipotent representations.

Other works : I have a short note [6] in which I give a proof of the analogue of Bezout's theorem for abelian varieties: any two subvarieties of complementary dimensions in a simple abelian variety intersect. When the paper was written, I did not know that the theorem was due to W. Barth, but the proof presented in [6] was different anyway.
The short note [26] to the paper of Schneider and Teitelbaum introduces the concept of locally algebraic representations, and suggestes an analogue of the Harish-Chandra sub-quotient theorem for $p$-adic representations of $p$-adic groups.
In paper [18] with Kumar Murty, we parametrise Tate cycles on products of two Hilbert modular surfaces in terms of Hilbert modular forms, including the precise information about the field of rationality.
L. Merel has proved an important theorem stating that the order of torsion on elliptic curves over a number field are bounded independent of the elliptic curve and the field, and depends only on the degree of the field. However, there are still no good bounds. In an attempt to see what might be the best bound, in a note with Yogananda [24], we estimate the bounds on torsion on CM elliptic curves.
I have made an analogue of a conjecture of Mazur on the density of rational points in the Euclidean topology on an Abelian variety to certain tori (isomorphic to $({\Bbb S}^1)^n$ but non-algebraic!), and proved it using the Schanuel conjecture in [29].
In a paper with C. Khare [30] we prove that an abstract homomorphism between the Mordell-Weil group of abelian varieties over a number field which respects reduction mod $p$, in fact arises from homomorphism of abelian varieties.
The paper [28] written with CS Rajan is a re-look at Sunada's theorem about isospectral Riemannian manifolds where we deduce it as a consequence of a simple lemma in group theory. In this paper we also conjecture, and verify in several cases, that the Jacobians of two Riemann surfaces with the same spectrum for Laplacian are isogenous (after an extension of the base field), and propose this as an Archimedean analogue of Tate's conjecture.
I have written some survey papers, of which [39], [40] might have some results which may not be found elsewhere.

Professional Recognition, Awards, Fellowships received :

 1. Sloan Fellowship at Harvard University 1988-89. 2. NSERC fellowship of the Canadian Government, 1993. 3. BM Birla Prize in Mathematics for the year 1994. 4. Elected fellow of the Indian Academy of Science in 1995. 5. Elected fellow of the National Academy of Science, India in 1997. 6. Swarna Jayanti Fellowship for Mathematics awarded in the year 98-99 for 5 years. 7. Shanti-Swarup Bhatnagar Award for Mathematical Sciences for the year 2002. 8. Ramanujan Award of the Indian Science Congress for the year 2005. 9. J.C. Bose fellowship 2010-2015.

Professional Experience:

 Research Scholar TIFR, Bombay 1980-1985 Graduate student Harvard University 1985-19 89 Research Assistant TIFR, Bombay 1989-1990 Fellow TIFR, Bombay 1990-1993 Reader TIFR, Bombay 1993-1997 Associate Professor Mehta Research Institute 1994- 1997 Professor Mehta Research Institute 1997- 2004 member Institute for Advanced Study Princeton, 1992-93 visitor University of Toronto 1993 visitor MSRI, Berkeley Spring 1995 visitor Harvard University Spring 1997 Visiting Associate Professor University of Chicago Spring 1998 Visiting Professor University of Chicago Spring 2000. Visiting Professor Cal. Tech. Spring 2003 member Institute for Advanced Study Princeton, 2006-07 Visitor University of California at San Diego, 2007-08 Research Professor MSRI Fall semester 2014

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