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Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of $SL_2(mathbb R)$, exhibiting it as the generator of $KK^1(mathbb C, mathfrak A)$, where $mathfrak A$ is the reduced $C^*$-algebra of the group. This yields a new and direct computation of the $K$-theory of $mathfrak A$. A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group. We provide a detailed analysis of the localised spectra of the Dirac operator and compute the Dirac cohomology.
We study the homotopy type of the space of the unitary group $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$ of the uniform Roe algebra $C^ast_u(|mathbb{Z}^n|)$ of $mathbb{Z}^n$. We show that the stabilizing map $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))
Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, sigma an elliptic automorphism of L leaving the form invariant, and A a sigma-invariant reductive subalgebra of L, such that the restriction of the fo
We propose novel asymptotically locally flat boundary conditions for Einstein Gravity without cosmological constant in four dimensions that are consistent with the variational principle. They allow for complex solutions that are asymptotically diffeo
We provide an explicit set of algebraically independent generators for the algebra of invariant differential operators on the Riemannian symmetric space associated with $SL_n(R)$.
We compute the overlap Dirac spectrum on three ensembles generated using 2+1 flavor domain wall fermions. The spectral density is determined up to $lambdasim$100 MeV with sub-percentage statistical uncertainty. The three ensembles have different latt