For any natural number n, the group $$G_n$$ of all invertible affine transformations n-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and left regular is a multiple this representation. We provide concrete realization $$\sigma $$ distinguished in two-dimensional case. explicitly decompose Hilbert $$L^2(G_2)$$ as direct sum left-invariant closed sub...

We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko’s in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x0 + · · · + x 2 n = 1} over the real numbers is just the...