Download Probabilities on the Heisenberg Group: Limit Theorems and by Daniel Neuenschwander PDF
By Daniel Neuenschwander
The Heisenberg workforce comes from quantum mechanics and is the best non-commutative Lie team. whereas it belongs to the category of easily hooked up nilpotent Lie teams, it seems that its precise constitution yields many effects which (up to now) haven't carried over to this greater category. This e-book is a survey of probabilistic effects at the Heisenberg staff. The emphasis lies on restrict theorems and their relation to Brownian movement. along with classical chance instruments, non-commutative Fourier research and sensible research (operator semigroups) is available in. The ebook is meant for probabilists and analysts drawn to Lie teams, yet given the various purposes of the Heisenberg team, it is going to even be worthwhile for theoretical phycisists really good in quantum mechanics and for engineers.
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Additional info for Probabilities on the Heisenberg Group: Limit Theorems and Brownian Motion
Example text
Proof: Suppose the functional equation holds. The functions eKgK certainly span H(G, K) over k. We have (eKgK ϕ)(h) = ϕ(hx) dx / meas (KgK) = KgK ϕ(hθ1 gθ2 ) dθ1 dθ2 / meas (K)2 = = K K ϕ(hθg) dθ / meas (K) = K since ϕ is right K-invariant. Then this is = ϕ(h)ϕ(g) = Λ(eKgK ) ϕ(h) by definition of ϕ and by the functional equation. Thus, Rη ϕ = Λ(η)ϕ Then Λ(η1 ∗ η2 )ϕ = Rη1 ∗η2 ϕ = = Rη1 Rη2 ϕ = Λ(η1 )Λ(η2 )ϕ Thus, ϕ is an eigenfunction and Λ is a k-algebra homomorphism, and then also ϕ(1) = 1. On the other hand, suppose that Λ is a k-algebra homomorphism and ϕ a normalized eigenvector.
Admissibility A smooth representation (π, K) of G is admissible if, for every compact open subgroup K of G the K-fixed vector space V K is finite-dimensional. This following assertion, and all further assertions in this section, follow directly from the ‘complete reducibility’ of smooth representations of compact totally disconnected groups. Proposition: An equivalent characterization of admissibility is that, for a fixed compact open subgroup K of G, every irreducible smooth representation δ of K has finite multiplicity in Res G K π.
A K-spherical representation of G is an irreducible (smooth) representation π of G with a (non-zero) k-spherical vector. Lemma: Let v = 0 be a K-spherical vector in a K-spherical representation π of G. Then πK = k · v Proof: As usual, let eK = chK / meas (K) The irreducibility of π implies that H(G) · v = π where H(G) is the full Hecke algebra. Therefore, for any w ∈ π K there is η ∈ H(G) so that ηv = w Then w = eK w = eK ∗ ηv = eK ∗ η ∗ eK v = Λ(eK ∗ η ∗ eK )v ∈ k · v ♣ This is the desired result.