Linear algebraic groups (11Exx, 14Lxx, 20Gxx), Division algebras (16Kxx, 16Hxx), Representations of finite groups (20Cxx)  and related structures, including Galois cohomology (12G05), Lie algebras (17Bxx), Jordan algebras (17Cxx) Compact groups (22Cxx), Lie groups (22Exx).

(with Anuradha Nebhani) Schur-Weyl duality for special orthogonal groups (submitted) (with U. Rehmann) Subfields of quaternion algebras (preprint) (with J. Oesterlé) Forms of unipotent groups (preprint) (with J. Oesterlé) On Gelfand models for finite Coxeter groups, J. Group Theory, 13:3, 429--439 (2010). [arXiv] Excellence of $G_2$ and $F_4$, Münster J. Math., 3, 213--220 (2010). [arXiv] On the orders of finite semisimple groups, Proc. Indian Acad. Sci. Math. Sci., 115:4, 411--427 (2005). [arXiv] Maximal tori determining the algebraic group, Pacific J. Math., 220:1, 69--85 (2005). [arXiv] Arithmetic of algebraic groups, Ph.D. thesis. [arXiv]

Linear algebraic groups are Zariski-closed subgroups of $GL_n$. In other words, these are the subgroups of $GL_n$ that are described by vanishing of certain polynomials. For example, the group of $n \times n$ matrices with determinant $1$, $SL_n$, is an algebraic group. A linear algebraic group $G$ is said to be reductive if it has no nontrivial connected normal subgroup containing only unipotent elements. Some of the important examples of reductive groups are $GL_n, SO_n, Sp_n$, and their products including the tori. A torus is an algebraic group that, over a separable closure, becomes isomorphic to a product of $GL_1$.    It is a natural to ask if a reductive group determined by the set of maximal tori contained in it. Clearly the question makes sense only over non-algebraically closed fields and in particular over fields that have many field extensions. Over such fields, this question is answered in the affirmative for all split groups, except that we cannot distinguish $B_n$ from $C_n$, in my main thesis paper (a copy in arxiv).    For non-split group, the answer to the above question is negative in general. However, it makes sense to ask an anologous question about division algebras. Are two quaternion division algebras that share the same set of maximal subfields isomorphic? This question was asked by Maneesh Thakur and appeared in print for the first time in a paper of Gopal Prasad and Andrei Rapinchuk. It can be easily seen that the answer to this question is affirmative over number fields and local fields. In a paper with Ulf Rehmann, it is proved that there are fields where the answer to the question can be as bad as possible.    Reductive algebraic groups over finite fields give rise to the finite groups of Lie type. These are also well-known for their connection to the finite simple groups. Emil Artin has proved that two finite simple groups of the same order are isomorphic except for some explicit pairs of finite simple groups, which we call counter-examples. All these counter-examples are given by finite groups of Lie type. Therefore it makes sense to ask the same question for finite reductive groups in general. This is discussed in my other paper during my Ph.D. (a copy in arxiv). As expected, there are many counter-examples for this questions also, however, we give some sort of geometric/topological reasoning for these counter-examples.    It has been the experience that split reductive groups are easier to study than the anisotropic groups which are the most difficult to study. To study anisotropic groups, Kersten and Rehmann introduced the notion of excellence of linear algebraic groups. In a short note, it is proved that groups of type $G_2$ and $F_4$ are excellent over any field.     A Gelfand model of a finite group $G$ is a complex linear representation that contains each of the irreducible representations of $G$ with multiplicity one. It is of interest to give ``natural'' construction of Gelfand models for a family of finite groups. Araujo and others gave a construction of such a model, called polynomial model, for certain Weyl groups. In a paper with Joseph Oesterle, we generalise their construction and give a uniform treatment for a finite Coxeter group, irreducible or otherwise (a copy in arxiv).

 

On excellence of $F_4$,      Mini-Cours: Formes quadratiques, Lens. FRANCE. (2006). Maximal tori determining algebraic groups,      Colloque Jeunes Chercheurs, Rennes. FRANCE. (2006). On the order of finite semisimple groups,      Colloquium, Indian Institute of Technology, Mumbai. (2005). Finite simple groups and their classification,   (These slides are very brief.)      VSRP Colloquium, TIFR, Mumbai. (2005). Maximal tori determining algebraic groups,      Conference and workshop on linear algebraic groups, quadratic forms and related topics, Eilat. ISRAEL. (2004). Maximal tori determining algebraic groups,      A session for young researchers in commutative algebra and algebraic geometry, IISc, Bangalore. (2003).

Faculty Member   (Currently Asso. Professor)   Postdoctoral Fellow   Postdoctoral Fellow   Postdoctoral Fellow   Research Scholar

Department of Mathematics, IITB, Mumbai, INDIA.       Fakultät für Mathematik, Bielefeld, GERMANY.   Institute de Mathématiques de Jussieu, Paris, FRANCE.   School of Mathematics, TIFR, Mumbai, INDIA.   Harish-Chandra Research Institute, Allahabad, INDIA.

2007 -- present       2007   2006   2004 -- 2006   1998 -- 2004

 

I have also been involved in organising following workshops. Steinberg festival, Nov 7--12, 2005. Algebraic groups: 2010, Dec 6--10, 2010.

$\vdots$ Emil Artin John Tate Benedict Gross Dipendra Prasad  Shripad M. Garge  (Find your genealogy tree here.)