Certain homomorphism spaces \({\mathcal H}_ K(G,U)\) and a theorem of unification on modular representations of Chevalley groups, finite and algebraic (Q1330055)

A theorem of unification is proved which explains the connection that links Hecke algebras with ordinary and modular representations of finite and algebraic Chevalley groups. The main result is the following: A theorem of unification: (i) Let \(G\) be a Chevalley group defined over an algebraically closed field \(K\) of characteristic \(p > 0\) and \(B\) be a Borel subgroup of \(G\) which is a semidirect product of \(U\) and \(H\). Let \({\mathcal H}={\mathcal H}_K(G,U)=(\text{ind}^G_U1)\cap{\mathcal S}(G/U)\) and \({\mathcal H}_\infty=\text{End}_{KG}(KG\otimes_{KU}1)\). Then all finite dimensional irreducible right \({\mathcal H}_\infty\)-submodules of \(\mathcal H\) are one- dimensional and afford rational linear characters of \(\mathcal H\), and there exists a one-one correspondence between the set of one-dimensional right \({\mathcal H}_\infty\)-submodules of \(\mathcal H\) and the equivalence classes of finite dimensional rational irreducible \(KG\)-modules such that \(\kappa(KG\otimes_{KU}1)\) is a finite dimensional rational irreducible \(KG\)-module of weight \(\lambda^{w_0}\) where \(K\kappa\) (\(\kappa\in {\mathcal H}\)) is a one-dimensional right \({\mathcal H}_\infty\)-submodule of \(\mathcal H\) affording a rational linear character \(\lambda^{w_0}\) of \(H\). (ii) Let \(G_0\) be a finite Chevalley group over a finite field \(F\) of characteristic \(p\) and of order \(q\). Let \({\mathcal H}_q=\text{End}_{KG_0}(1^{G_0}_{U_0})\). Then all irreducible right \({\mathcal H}_q\)-modules are one-dimensional and the socle of \({\mathcal H}_q\) is multiplicity-free, and there exists a one-one correspondence between the set of one-dimensional right \({\mathcal H}_q\)-submodules of \({\mathcal H}_q\) and the equivalence classes of irreducible \(KG_0\)- modules such that \(\kappa(1^{G_0}_{U_0})\) is an irreducible \(KG_0\)-module of weight \((\psi|_{\{A_h\mid h\in H_0\}},\psi(A_{w_1}),\dots,\psi(A_{(w_n)}))\) where \(K\kappa\) (\(\kappa\in{\mathcal H}_q\)) is a one-dimensional right \({\mathcal H}_q\)- submodule of \({\mathcal H}_q\) affording a linear character \(\psi\) of \({\mathcal H}_q\).