Shripad M. Garge


Research Summary:

I work on the rationality properties of algebraic groups.

Let $H$ be a connected, reductive algebraic group defined over a field $k$. It is natural to ask whether $H$ is determined by the set of $k$-isomorphism classes of maximal $k$-tori in it. I studied this question over global fields, local non-archimedean fields and finite fields. I prove following theorems in the preprint [1].

Theorem 1.1. Let $k$ be a finite field, a global field or a local non-Archimedean field. Let $H_1$ and $H_2$ be two split, connected, reductive algebraic groups defined over $k$. Suppose that for every maximal $k$-torus $T_1 \subset H_1$ there exists a maximal $k$-torus $T_2 \subset H_2$, such that the tori $T_1$ and $T_2$ are $k$-isomorphic and vice versa. Then the Weyl groups $W(H_1)$ and $W(H_2)$ are isomorphic.

Moreover, if we write the Weyl groups $W(H_1)$ and $W(H_2)$ as a direct product of the Weyl groups of simple algebraic groups, $W(H_1) = \prod_{\Lambda_1} W_{1, \alpha}$, and $W(H_2) = \prod_{\Lambda_2} W_{2, \beta}$, then there exists a bijection $i: \Lambda_1 \rightarrow \Lambda_2$ such that $W_{1, \alpha}$ is isomorphic to $W_{2, i(\alpha)}$ for every $\alpha \in \Lambda_1$.

Theorem 1.2. Let $k$ be as in the previous theorem. Let $H_1$ and $H_2$ be two split, connected, semisimple algebraic groups defined over $k$ with trivial center. Write $H_i$ as a direct product of simple groups, $H_1 = \prod_{\Lambda_1} H_{1, \alpha}$, and $H_2 = \prod_{\Lambda_2} H_{2, \beta}$. If the groups $H_1$ and $H_2$ satisfy the condition given in the above theorem, then there is a bijection $i: \Lambda_1 \rightarrow \Lambda_2$ such that $H_{1, \alpha}$ is isomorphic to $H_{2, i(\alpha)}$, except for the case when $H_{1, \alpha}$ is a simple group of type $B_n$ or $C_n$, in which case $H_{2, i(\alpha)}$ could be of type $C_n$ or $B_n$.

My current work concerns finite groups of Lie type. It is a theorem of Emil Artin, Tits, Kimmerle et al. that a finite simple group is determined by its order except for the following exceptions:

\begin{displaymath}\big(A_3(2), A_2(4)\big) {\rm ~and~} \big(B_n(q), C_n(q)\big) {\rm ~for~} n \geq 3, q {\rm ~odd.}\end{displaymath}

I am trying to extend this theorem to the finite semisimple groups of Lie type. Following results are expected.

  1. Let $H_1$ and $H_2$ be two semisimple simply connected groups defined over finite fields ${\Bbb F}_{q_1}$ and ${\Bbb F}_{q_2}$. Suppose that neither of the $q_1, q_2$ is in the set $\big\{2, 3, 4, 5, 7, 8, 9, p, 2^s\big\}$ where $p$ is a prime of the form $2^r \pm 1$ and $2^s + 1$ is a prime. If the order of $H_1({\Bbb F}_{q_1})$ is the same as the order of $H_2({\Bbb F}_{q_2})$, then $q_1 = q_2$.

  2. Let $H_1$ and $H_2$ be two semisimple simply connected algebraic groups defined over a finite field ${\Bbb F}_q$ such that the order of $H_1({\Bbb F}_q)$ is the same as the order of $H_2({\Bbb F}_q)$, then the orders of $H_1({\Bbb F}_{q^n})$ and $H_2({\Bbb F}_{q^n})$ are the same for all $n$.

I also hope to characterize the semisimple simply connected groups $H_1, H_2$ defined over a finite field ${\Bbb F}_q$, which have exactly two simple factors and such that the order of $H_1({\Bbb F}_{q})$ is equal to the order of $H_2({\Bbb F}_{q})$.

Preprints:

  1. Maximal tori determining algebraic groups.
  2. On the order of finite semisimple groups (in preparation).

Conference/Workshops Attended:

  1. Instructional Workshop and International Conference on Geometric Group Theory, Indian Institute of Technology, Guwahati. (December 2 - 21, 2002).
  2. Workshop on the Computational Aspects of Algebraic Geometry, Harish-Chandra Research Institute, Allahabad. (January 1 - 11, 2003).

Visits to other Institutes:

  1. Tata Institute of Fundamental Research, Mumbai. (May 1 - July 31, 2002 and February 28 - March 28, 2003).
  2. Bhaskaracharya Pratishthana, Pune. (March 31 - April 15, 2003).

Invited lectures/Seminars:

  1. Lectured in the International Conference on Geometric Group Theory at IIT, Guwahati, on ``Conjugacy of Weyl Groups''.
  2. Gave a series of lectures on ``Central Simple Algebras and the Brauer Group'' at the Bhaskaracharya Pratishthana, Pune.


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