Present Position
Professor, Department of Mathematics, IIT Bombay.
PhD from Harvard University, 1989.
Mathematical Interest
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Editor: Journal of Number Theory.
Editor: Math. Zeitschrift.
Editor: Journal of Ramanujan Mathematical Society.
Editor: Proceedings of Indian Academy.
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.
(Electronic Journal) Representation Theory, vol. 5, 111-128 (2001). geometry, and number theory, 699--709, Johns Hopkins Univ. Press, Baltimore, MD, 2004. Contemporary
Math., > vol 478, AMS, pp. 99-101.
in the representation theory of classical groups; vol. 346, pp 1-109, Asterisque (2012)> .
the memory of Prof. Hiroshi Saito, vol. 208, pp. 171-199 (2012).>
by coefficients restricted to quadratic subfields by M. Krishnamurthy;''
Journal of Number Theory > , Volume 132, Issue 6, Pages 1359-1384 (June 2012)
``Representation theory,
Number theory, and Invariant theory,'' Progress in Mathematics, volume 323 (2017) > pages 1-22.
Papers in Conference Proceedings
Jan 01, 2016 to June 30, 2016: Chaire Morlet at CIRM and Aix-Marseille Université.
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Managing Editor: International Journal of Number Theory 2005-2010.
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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. |
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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