arXiv subjects

Sean Rogers

Publications and source records attributed to Sean Rogers.

2 recordsLinked to original sources

arXiv2019

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ ξ_{n,k}: Rep^\mathbb{C}_{ω_1-poly}(Sp(2k)) \to H^*(IG(n-k, 2n), \mathbb{C})$. Our main result asserts that $ ξ_{n,k}$ is injective when n tends to $\infty$ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

math.RT↗

arXiv2017

An Explicit Determination of the Springer Morphism

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $ρ:G\rightarrow GL(V_λ)$ be an irreducible representation of highest weight $λ$. Suppose that $ρ$ has finite kernel. Springer defined adjoint-invariant regular map with Zariski dense image from the group to its Lie algebra, $θ_λ:G\rightarrow\mathfrak{g}$, which depends on $λ$ [Kumar]. By a lemma in Kumar's recent paper, $θ_λ$ takes the maximal torus to its Lie algebra $\mathfrak{t}$. Thus, for a given simple group $G$ and an irreducible representation $V_λ$, one may write $θ_λ(t)=\sum\limits_{i=1}^n c_i(t)\check{α_i}$, where the simple co-roots $\{\check{α_i}\}$ are a basis for $\mathfrak{t}$. We give a complete determination of these coefficients $c_i(t)$ for any simple group $G$ as a sum over the weights of the torus action on $V_λ$.

math.RT↗