Shripad M. Garge

Research Summary:

I work on the arithmetic of linear algebraic groups. This involves understanding the structure of linear algebraic groups over certain special fields, like global fields, local fields and finite fields.

One of the questions that I studied in this year concerned with the structure of linear algebraic groups over finite fields. Let H be a semisimple linear algebraic group defined over a finite field F_q. It is a theorem of Emil Artin and Tits et. al., that if H is a simple algebraic group, then it is determined (upto isomorphism) by the order of H(F_q), the F_q rational points of H, except in some cases which can be explicitly given. It is natural to study the situation for semisimple groups. The following theorems are proved in [2].

Theorem 1 Let H₁ and H₂ be two split semisimple simply connected algebraic groups defined over finite fields F_q1 and F_q2 respectively. Let   X denote the set   { 8, 9, 2^r, p }, where 2^r + 1 is a prime and p is a prime of the form 2^s ± 1. Suppose that, for i = 1, 2; A₁ is not one of the direct factors of H_i whenever q_i belong to X and B₂ is not a direct factor of H_i whenever q_i = 3. Then, if |H₁(F_q1)| = |H₂(F_q2)|, the characteristics of F_q1 and F_q2 are the same.

Theorem 2 H₁ and H₂ be two split semisimple simply connected algebraic groups defined over finite fields F_q1 and F_q2 of the same characteristic. Suppose that the order of the finite groups H₁(F_q1) and H₂(F_q2) are the same, then q₁ = q₂. Moreover the fundamental degrees (and the multiplicities) of the Weyl groups W(H₁) and W(H₂) are the same.

Thus, the situation is reduced to understanding the groups H₁ and H₂ defined over a finite field F_q such that |H₁(F_q)| = |H₂(F_q)|. We put an equivalence structure on the set of pairs of groups of the same order defined over F_q given by (H₁, H₂) &sim (H₁', H₂') if and only if there exist groups G, G' defined over the same field such that

H₁×G isomorphic to H₁'×G' and H₂×G isomorphic to H₂'×G' .

The set of equivalence classes then admits the structure of an abelian group in an obvious way. We give an explicit set of generators for this group. We also give a geometric reasoning for this coincidence of orders.

The other question that I studied in this year is about excellence of algebraic groups.

An algebraic group defined over a field k is said to be excellent if for any extension L/k, the anisotropic kernel of G &otimes_k L is defined over k. This notion is a generalization of the notion of excellence in the theory of quadratic forms. The excellence properties of some groups of classical type have been studied so far. These properties are usually related with the arithmetic properties of the field k. However, the exceptional group of type G₂ is excellent over any field. We prove that the group of type F₄ is also excellent over any field [3]. The proof uses the theory of Albert algebras and the octonion algebras in detail.

Preprints:

1. Maximal tori determining the algebraic group (to appear in the Pacific Journal of Mathematics)
2. On the order of finite semisimple groups (submitted)
3. Excellence properties of F₄.

Conference/Workshops Attended:

1. International conference on Algebra and Number theory, Hyderabad (11--16 December, 2003).
2. A session for young researchers in commutative algebra and algebraic geometry, IISc, Bangalore (December 16, 2003).
3. Joint India-AMS mathematics meeting, IISc, Bangalore (17--20 December, 2003).
4. International colloquium on algebraic groups and homogeneous spaces, TIFR, Mumbai (6--14 January, 2004).
5. Conference and workshop on linear algebraic groups, quadratic forms and related topics, Eilat, Israel (1--5 February, 2004).

Visits to other Institutes:

1. Tata Institute of Fundamental Research (December 25, 2003 -- January 25, 2004).
2. University of Tel Aviv, Israel (February 6--15, 2004).

Invited lectures/Seminars:

1. Maximal tori determining the algebraic group, A session for young researchers in commutative algebra and algebraic geometry, IISc, Bangalore.
2. Maximal tori determining the algebraic group, Conference and workshop on linear algebraic groups, quadratic forms and related topics, Eilat, Israel.