The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite-dimensional l-adic representations of Gal(Q/K) for some number field K. We exhibit examples (with K = Q) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of th...

The `-adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite-dimensional `-adic representations of Gal(Q/K) for some number field K. We exhibit examples (with K = Q) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of th...

In this article we define a new family of subgroups of Fuchsian groups H( √ m), for a squarefree positive integer m, and calculate their index in H( √ m) and their parabolic class number. Moreover, we will show that the index of these subgroups is closely related to the solvability of a quadratic congruence x2 ≡ m(mod n) and the number of inequivalent solutions of a quadratic congruence x2 ≡ 1(...

1. Spaltenstein’s theorem and Hotta’s construction 1 2. Equivariant cohomology and divided differences 5 3. Review of: Borel subgroups, parabolic subgroups, the Bruhat decomposition 10 4. The Steinberg scheme 12 5. Algebras of constructible correspondences 13 6. Hall algebras 17 7. Geometric construction of Uq(n+) for simply-laced Lie algebras 20 8. Convolution in Borel-Moore homology 23 9. Gro...

Abstract We identify the exit path $\infty $-category of reductive Borel–Serre compactification as nerve a $1$-category defined purely in terms rational parabolic subgroups and their unipotent radicals. As immediate consequences, we fundamental group compactification, recovering result Ji–Murty–Saper–Scherk, obtain combinatorial incarnation constructible complexes sheaves on elements derived fu...

We give a necessary and sufficient condition for pair of parabolic subgroups $P$ $Q$ $G=GL_n(\mathbb{R})$ such that the tensor product any two unitarily induced representations from are tempered. also an $L^p$-estimate matrix coefficients regular on $L^2(G/L)$ when $L$ is Levi subgroup $G$.

We study the properties of groups that have presentations in which generating set is a fixed involutions and all additional relations are even length. consider parabolic subgroups such group show every element has factorization with respect to given subgroup. Furthermore, we give counterexample, using cluster presentation, demonstrates this not necessarily unique.