We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a ...

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid G is equivalent to the continuity of the implied convolution product f * g for all pairs of functions / and g. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual C*-algebra associated with a Haar measure on G cannot, in general, be constr...

We show that the empirical eigenvalue measure for sum of d independent Haar distributed n-dimensional unitary matrices, converge for n → ∞ to the Brown measure of the free sum of d Haar unitary operators. The same applies for independent Haar distributed n-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes t...

In this short note, we have given a short proof for the existence of the Haar measure on commutative locally compact hypergroups based on functional analysis methods by using Markov-Kakutani fixed point theorem.

If C ≃ 2 is the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from C into [0, 1] sends Haar measure to Lebesgue measure on the interval. In fact, C admits many distinct topological group structures. In this note, we show that the Haar measures induced by these distinct group structures all share this property. We prove this by showin...

This informal set of notes contains some of the new results which will appear in the author’s 2013 dissertation. We show that every infinite product of locally compact non-compact groups decomposes into the disjoint union of a Haar null set and a meager set, which gives a partial positive answer to a question of Darji. We also show that the compact sets in each such product group are always Haa...

We compute the Riemannian volume of the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten’s volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten’s volume formula for the moduli space of flat connections on a Riemann surface using Haar measure. ———————–

Let K be a (commutative) locally compact hypergroup with a left Haar measure. Let L1(K) be the hypergroup algebra of K and UCl(K) be the Banach space of bounded left uniformly continuous complex-valued functions on K. In this paper we show, among other things, that the topological (algebraic) center of the Banach algebra UCl(K)* is M(K), the measure algebra of K.