By Michael Cowling, Edward Frenkel, Masaki Kashiwara, Alain Valette, David A. Vogan, Nolan R. Wallach, Enrico Casadio Tarabusi, Andrea D'Agnolo, Massimo Picardello

Six prime specialists lecture on a large spectrum of contemporary effects just about the name, supplying either a pretty good reference and deep insights on present study job. Michael Cowling provides a survey of varied interactions among illustration thought and harmonic research on semisimple teams and symmetric areas. Alain Valette remembers the concept that of amenability and indicates the way it is utilized in the evidence of stress effects for lattices of semisimple Lie teams. Edward Frenkel describes the geometric Langlands correspondence for advanced algebraic curves, targeting the ramified case the place a finite variety of common singular issues is permitted. Masaki Kashiwara stories the connection among the illustration idea of actual semisimple Lie teams and the geometry of the flag manifolds linked to the corresponding advanced algebraic teams. David Vogan bargains with the matter of having unitary representations out of these bobbing up from advanced research, resembling minimum globalizations discovered on Dolbeault cohomology with compact aid. Nolan Wallach illustrates how illustration concept is said to quantum computing, targeting the learn of qubit entanglement.

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**Additional info for Representation theory and complex analysis: CIME summer school, 2004**

**Sample text**

This is a reformulation of a celebrated result of A. Selberg (generalising a result of G. Roelcke for SL(2, R)/SL(2, Z) which states, in our language, that for this choice of Γ , the representation π0 satisﬁes a L2+ estimate). Selberg also conjectured that π0 satisﬁes an L2+ estimate. This result is usually phrased in terms of the ﬁrst nonzero eigenvalue of the Laplace–Beltrami operator on the space K\G/Γ , a quotient of the hyperbolic upper half plane; the representation theoretic version is due to I.

1997. 33. M. Cowling, Measure theory and automorphic representations,, to appear. Bull. Kerala Math. Assoc. 3 (Special issue) (2006), 139–153. 34. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math.

5. Let (Ht )t>0 be the heat semigroup. Then the following hold: (i) for all p in [1, ∞], |||Ht |||p = exp −(1 − δ(p)2 )bt ∀t ∈ R+ ; Applications of Representation Theory to Harmonic Analysis 19 (ii) for all p, q such that 1 ≤ p ≤ q ≤ ∞, |||Ht |||p;q ∼ t−n(1/p−1/q)/2 ∀t ∈ (0, 1]; (iii) for all p, q such that either 1 ≤ p < q = 2 or 2 = p < q ≤ ∞, |||Ht |||p;q ∼ t−ν/4 exp(−bt) ∀t ∈ [1, ∞); (iv) for all p, q such that 1 ≤ p < 2 < q ≤ ∞, |||Ht |||p;q ∼ t−ν/2 exp(−bt) ∀t ∈ [1, ∞); (v) for all p, q such that 1 ≤ p < q < 2, |||Ht |||p;q ∼ t− /2q exp −(1 − δ(q)2 )bt ∀t ∈ [1, ∞); (vi) for all p, q such that 2 < p < q ≤ ∞, |||Ht |||p;q ∼ t− /2p exp −(1 − δ(p)2 )bt ∀t ∈ [1, ∞).