### Citations

516 | groups, Lie algebras and their representation - Varadarajan - 1984 |

277 | Lie group representations on polynomial rings, - Kostant - 1963 |

223 |
Representation Theory of Semisimple Groups: An Overview Based on Examples,
- Knapp
- 1986
(Show Context)
Citation Context ...s from [O, Corollary 2.9] that K̂qsp = K̂sp = {Ctriv, s+, s−} = {0,±2}. Let α ∈ a∗ be such that Σ+ = {α} (and hence R+ = {2α}). The classification of the irreducible (gC, K)-modules is classical (cf. =-=[Kn]-=-). Here we fix some notation. For n = 1, 2, . . . we put EnG = the irreducible representation with dimension n, Dn,+G = the discrete series representation with K-types {n + 1, n+ 3, . . .}, Dn,−G = th... |

147 | Spherical functions on a semisimple Lie group. - Harish-Chandra - 1958 |

142 |
Opdam E. M.: Root systems and hypergeometric functions
- Heckman
- 1987
(Show Context)
Citation Context ...sawa decomposition G = NAK we have G/K ≃ NA. Let γ0 : C ∞(G/K) −→ C∞(A) be the natural restriction map and W the Weyl group for (g, a). Then the second property in (1.2) implies γ0(φλ) ∈ C ∞(A)W . By =-=[HO]-=-, Heckman and Opdam started their studies on the systems of hypergeometric differential equations, which are certain modification of the system of differential equations satisfied by γ0(φλ). At the ea... |

140 |
Harmonic analysis for certain representations of graded Hecke algebras
- Opdam
- 1995
(Show Context)
Citation Context ... S(aC) be as in Proposition 3.3 (iii). Then (4.2) Tk(La) = ∂(La) + ∑ α∈R+ k(α) ( coth α 2 ∂(Hα)− |α|2 4 sinh2 α 2 (1− sα) ) +B(Hρk, Hρk). Proof. Proposition 4.3 (i), (ii) are given in [Ch1] (see also =-=[Op1]-=-). (iii) is calculated in [Sha]. Remark 4.4. If k = m then ρk = ρ, k1 =m1 and Hk1 = H. NON-INVARIANT RADIAL PART FORMULAS 15 5. Radial part formula, I The next theorem is a generalization of (1.3) t... |

115 |
Orbits and representations associated with symmetric spaces,
- Kostant, Rallis
- 1971
(Show Context)
Citation Context ... Cartan decomposition of g. If S denotes the image of the symmetrization map of S(sC) then one has PG(Ctriv) ≃ S ⊗ vtriv. Thus a K-type of PG(Ctriv) is a K-type of S(sC) and hence is belonging to K̂M =-=[KR]-=-. The other conditions follow from Theorem 6.5 (i), (ii) and Proposition 6.3. Proposition 8.11. Suppose M = (MG,MH) ∈ Cw-rad satisfies (rad-2) and MH has a W -invariant element φW . Fix a non-zero v... |

94 |
Interwining operators for semi-simple groups
- Knapp, Stein
- 1971
(Show Context)
Citation Context ...kes sense when −Reλ is sufficiently dominant. (By [Sch] the integral is convergent when λ satisfies Reλ(α∨) < 0 for all α ∈ Σ+ ∩ −w−1Σ+.) This operator clearly does not depend on the choice of w̄. In =-=[KnS1]-=- Knapp and Stein prove that AG(w, λ), as an operator acting on C∞(K/M) ≃ BG(λ) with the holomorphic parameter λ, extends meromorphically in λ to the whole a∗ C . Now let us assume for each α ∈ Π the H... |

77 | Cuspidal local systems and graded Hecke algebras I,
- Lusztig
- 1988
(Show Context)
Citation Context ...ii). 4. Graded Hecke algebras and Cherednik operators Let Π be the system of simple roots in R1 = 2Σ \ 4Σ corresponding to the positive system R+1 = 2Σ + \ 4Σ+. Definition 4.1 (graded Heck algebras =-=[Lu]-=-). Let k : W\R1 → C be a multiplicity function. Then there exists uniquely (up to equivalence) an algebra Hk over C with the following properties: (i) Hk ≃ S(aC)⊗ CW as a C-linear space; (ii) The maps... |

74 |
D u nk l, Differential-difference operators associated to reflection groups,
- F
- 1989
(Show Context)
Citation Context ...nsider the infinitesimal (or “tangential”) version of radial part formulas. Let s be the −1-eigenspace of θ in g. For the Cartan motion group GCM := K⋉s and the rational Dunkl operators introduced by =-=[Dun]-=-, we have similar results to the case of G/K. Using the K-module isomorphism (1.14) C∞(G/K) ∼−→ C∞(s) ; f 7−→ f(exp ·). and the W -module isomorphism (1.15) C∞(A) ∼−→ C∞(a) ; ϕ 7−→ ϕ(exp ·), we identi... |

56 | A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups - Heckman - 1989 |

38 |
and Geometric Analysis
- Groups
- 1984
(Show Context)
Citation Context .../K); r(∆)f = γ(∆)(λ)f for ∆ ∈ U(gC) K } has a role of constitutional unit of C∞(G/K). This G-module is known as the solution space for a maximal system of invariant differential operators on G/K (cf. =-=[Hel4]-=-). As a direct link between A (A, λ) and A (G/K, λ), we have a linear bijection: (1.6) γ0 : A (G/K, λ) ℓ(K) ∼−→ A (A, λ)W ; φλ 7−→ γ0(φλ). This comes from the Chevalley restriction theorem, Harish-Cha... |

36 | Duality between sl(n,C) and the degenerate affine Hecke algebra, - Arakawa, Suzuki - 1998 |

32 |
Integrable connections related to zonal spherical functions
- Matsuo
- 1992
(Show Context)
Citation Context ..., U = {H ∈ a; |α(H)| < 2π for any α ∈ R}, U+ = U ∩ a+, (a+ iU)reg = {H ∈ a+ iU ; α(H) 6= 0 for any α ∈ R}. NON-INVARIANT RADIAL PART FORMULAS 113 Definition A.3 (the Knizhnik-Zamolodchikov connection =-=[M]-=-). Let E = (a+iU)×CW be the trivial vector bundle with fiber CW over the complex manifold a + iU . The Knizhnik-Zamolodchikov connection ∇ = ∇(λ,k) is a connection on (a+iU)reg×CW ⊂ E whose covariant ... |

28 | The sum of generalized exponents and Chevalley’s restriction theorem for modules of covariants, - Broer - 1995 |

23 |
A unification of Knizhnik-Zamolodchikov equations and Dunkl operators via affine Hecke algebras
- Cherednik
- 1991
(Show Context)
Citation Context ...tion of the system of differential equations satisfied by γ0(φλ). At the early stage the existence of such modification had been quite non-trivial. But after a while Cherednik operators introduced by =-=[Ch1]-=- turned out to provide an elegant method to construct the modified systems. (This idea is due to [Hec2], in which Heckman operators play the same role as Cherednik operators.) In this context, a key f... |

12 |
Radon-Fourier transform on symmetric spaces and related group representations
- Helgason
- 1965
(Show Context)
Citation Context ...es. Of course they constitute a morphism in Crad (Theorem 13.5). In §14 we study the relation between the Helgason-Fourier transform FG and the Opdam-Cherednik transform FH introduced respectively in =-=[Hel1]-=- and [Op1]. The Paley-Wiener theorems, the inversion formulas and the Plancherel formulas for both transforms can be successfully combined in Crad (Theorem 14.16). In §15 we prove the generalized Chev... |

12 | reductive groups. I, Pure and Applied Mathematics 132 - Wallach, Real - 1988 |

11 |
Jeu, “Paley–Wiener theorems for the Dunkl transform
- de
(Show Context)
Citation Context ...→ S(aC), which is also denoted by γ0. For X ∈ s let ∂(X) denote the X-directional derivative operator on s. Extend the linear map ∂ : s→ EndC C∞(s) to an algebra homomorphism ∂ : S(sC)→ EndCC∞(s). In =-=[Je]-=- de Jeu gives a simple proof for the fact that the Dunkl operator D : S(aC) → EndCC ∞(a) with a special parameter (Definition 3.1) satisfies a radial part formula (1.16) γ0 ( ∂(∆)f ) = D ( γ0(∆) ) γ0(... |

6 | Generalization of Harish–Chandra’s basic theorem for Riemannian symmetric spaces of non-compact type,
- Oda
- 2007
(Show Context)
Citation Context ...ple, (1.3) holds for any ∆ ∈ U(gC)K and any K-finite f ∈ C∞(G/K) such that all K-types in ℓ(U(kC))f are single-petaled (Theorem 5.1). A single-petaled K-type is a special kind of K-type introduced by =-=[O]-=-. We denote the set of singlepetaled K-types by K̂sp (Definition 2.1). In [Op1] Opdam studies the following system of differential-difference equations: (1.5) T (∆)ϕ = ∆(λ)ϕ ∀∆ ∈ S(aC) W . Here λ ∈ a∗... |

5 | A Lie-theoretic construction of some representations of the degenerate affine and double affine Hecke algebras of type BCn, Represent. Theory 13 - Etingof, Freund, et al. - 2009 |

5 | notes on Dunkl operators for real and complex reflection groups, AMS Memoire 8 - Lecture - 2000 |

4 | Trapa, Functors for unitary representations of classical real groups and affine Hecke algebras - Ciubotaru, E |

4 |
Intégrales d’entrelacement et fonctions de Whittaker.", Bull.Soc.Math.de France 99
- Schiffmann
- 1971
(Show Context)
Citation Context ...∈ a∗ C , [KuS] shows the intertwining operator AG(w, λ) : BG(λ)→ BG(wλ) formally given by AG(w, λ)F (g) = ∫ w̄−1Nw̄∩ θN F (gw̄n̄) dn̄ converges and makes sense when −Reλ is sufficiently dominant. (By =-=[Sch]-=- the integral is convergent when λ satisfies Reλ(α∨) < 0 for all α ∈ Σ+ ∩ −w−1Σ+.) This operator clearly does not depend on the choice of w̄. In [KnS1] Knapp and Stein prove that AG(w, λ), as an opera... |

2 |
Uniformly bounded representations. III. Intertwining operators for the principal series on semisimple groups
- Kunze, Stein
- 1967
(Show Context)
Citation Context ... 11.1 (ii). Hence Ξ♮(DH(Ctriv, λ̄)) ⊂ U(wλ) = DG(Ctriv, λ̄). 13. Intertwining operators For an arbitrary w ∈ W let w̄ ∈ NK(a) be its lift. Let dn̄ be a Haar measure of w̄−1Nw̄ ∩ θN . For λ ∈ a∗ C , =-=[KuS]-=- shows the intertwining operator AG(w, λ) : BG(λ)→ BG(wλ) formally given by AG(w, λ)F (g) = ∫ w̄−1Nw̄∩ θN F (gw̄n̄) dn̄ converges and makes sense when −Reλ is sufficiently dominant. (By [Sch] the inte... |

1 |
Etude analytique et probabiliste de laplaciens associés à des systèmes de racines: laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines
- Schapira
- 2006
(Show Context)
Citation Context ... (iii). Then (4.2) Tk(La) = ∂(La) + ∑ α∈R+ k(α) ( coth α 2 ∂(Hα)− |α|2 4 sinh2 α 2 (1− sα) ) +B(Hρk, Hρk). Proof. Proposition 4.3 (i), (ii) are given in [Ch1] (see also [Op1]). (iii) is calculated in =-=[Sha]-=-. Remark 4.4. If k = m then ρk = ρ, k1 =m1 and Hk1 = H. NON-INVARIANT RADIAL PART FORMULAS 15 5. Radial part formula, I The next theorem is a generalization of (1.3) to some cases where f is no long... |