Let G be a connected linear algebraic group defined over a field k. Serre's injectivity question asks whether a principal homogeneous space under G admitting a zero cycle of degree one has a k-rational point. This question has been answered in the affirmative by Jodi Black for absolutely simple simply connected or adjoint k-groups of classical type. In this talk, using Merkurjev-Barquero's norm principle, we extend Jodi's result to reductive groups with Dynkin diagrams containing only connected components of type A_n, B_n or C_n. We also discuss the (non-trialitarian) type D_n case and give a scalar obstruction defined up to spinor norms whose vanishing will imply norm principle and yield a positive answer to Serre's question.
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C}$. The intersection of a Schubert variety (closure of an Orbit for the Borel subgroup of $G$ ) and an opposite Schubert variety ( Closure of an Orbit for the Opposite Borel subgroup ) is known as a Richardson variety. Since it is invariant under the action of a maximal torus, it is interesting to study its quotients. In this talk we prove that every power of the projective line can be obtained as such quotients.
Quadratic forms may admit very special properties in a few low dimensions. We describe examples which are standing out in Sridharan's work : exceptional isomorphisms, composition of forms and triality.
Quadratic forms over fields of characterstic 2 have usually been studied as isolated objects. In this talk we present an application of quadratic forms in characterstic 2 to explicitly write the Wedderburn decomposition of rational group algebras of certain 2-groups. This is a joint work with Dilpreet Kaur.
Let $G$ be a connected linear semsimple group and $K$ be (connected) closed subgroup such that AD($K$) is a maximal compact subgroup of AD($G$). Let $\mathfrak g$ be the Lie algebra of $G$ and $\,\mathfrak k$ the Lie sibalgebra of $\mathfrak g$ corresponding to $K$. Let $\rho$ be an irreducible unitary representation of $G$ on a separable complex Hilbert space $\math cal H_{\rho}$ and let $H_{\rho}$ the space of $K$-finite vectors in $\mathcal H_{\rho}$ considered as a $(\mathfrak g$-$K)$-module. It is a theorem of Casselmanand Millicic that $H_{\rho}$ admits an imbeddig as a $(\mathfrak g$-$K)-module in a principal series module. We give a new proof of this theorem in the special case when $\mathfrak g$ has ($\R$-) rank 1. In fact we prove a somewhat stronger result in this special case, viz. that any irreducible $(\mathfrak g$-$K)$-module $M$ is imbeddable in a principal series representation - we do not assume that the module is one obtained from a unitary representation of $G$. As a consequence one concludes that an irreducible $(\mathfrak g$-$K)$-module is necessarily a Harish-Chandra-module, i.e, the $K$-isotypical components of $M$ are finite dimensional. That $H_{\rho}$ is an irreducible $(\mathfrakg$-$K)$ module whic has this last property is a theorem of Harish-Chandra.
We discuss R-equivalence classes of some adjoint classical groups over the function field F of a smooth geometrically integral curve over a p-adic field (with p not 2).
Let $(Q, q)$ be a inner product space over a commutative ring $R$, and consider the Dickson-Siegel-Eichler-Roy's subgroup of the orthogonal group $O_R(Q \perp H(R)^n)$, $n \geq 1$. We show that it is a normal subgroup of $O_R(Q \perp H(R}^n)$, for all $n$, except when $n = 2$. This is a joint work with A.A. Ambily.