##
**Lecture notes on Dunkl operators for real and complex reflection groups.**
*(English)*
Zbl 0984.33001

MSJ Memoirs. 8. Tokyo: Mathematical Society of Japan, viii, 90 p. (2000).

These memoirs are based on a series of lectures presented by the author at RIMS, Kyoto University, in 1997. It is divided in two parts. The first part deals with the spectral theory of Dunkl-Cherednik operators and outlines the most recent developments in the theory of hypergeometric functions associated with root systems. The second part studies the fake degrees of complex reflection groups and a topological cyclotomic Hecke algebra constructed by the monodromy representation of a certain system of differential operators.

In the late 80ies, G.J. Heckman and E.M. Opdam defined the hypergeometric functions associated with root systems as analytic continuation of the Harish-Chandra’s spherical functions on Riemannian symmetric spaces. See G. J. Heckman and E. M. Opdam [Compos. Math. 64, 329-352 (1987; Zbl 0656.17006)], G. J. Heckman [ibid. 64, 353-373 (1987; Zbl 0656.17007)] and E. M. Opdam [ibid. 67, 21-49, 191-209 (1988; Zbl 0669.33007, Zbl 0669.33008)]. The analytically continued parameters are the root multiplicities. The definition of Heckman-Odpam’s hypergeometric functions is based on a generalization of the system of differential equations satisfied by Harish-Chandra’s spherical functions. The system represents the common spectral theory of a commuting family of differential operators, the so-called hypergeometric differential operators. In the symmetric case, they agree with the radial components on a maximal split Cartan subgroup of the invariant differential operators on the given Riemannian symmetric space. The construction of the hypergeometric differential operators became of completely algebraic nature when G. J. Heckman [Invent. Math. 103, 341-350 (1991; Zbl 0721.33009)] obtained them using a certain family of first-order non-commuting differential-reflection operators as building blocks. Heckman’s algebraic construction was later improved by I. Cherednik [Invent. Math. 106, No. 2, 411-432 (1991; Zbl 0742.20019)] with the introduction of another family of first-order differential-reflection operators which is moreover commutative. These operators are nowadays known as Dunkl-Cherednik operators. The harmonic analysis of the eigenfunctions of the Dunkl-Cherednik operators has been developed by E. M. Opdam in [Acta Math. 175, No. 1, 75-121 (1995; Zbl 0836.43017)].

To give a more detailed description of the first part of these lecture notes, let us fix some notation. Let \(\mathfrak a\) denote the Euclidean space with complexification \(\mathfrak h\). Let \(R \subset \mathfrak a^*\) be a root system with Weyl group \(W\), and let \(k=(k_\alpha)\) be a multiplicity function on \(R\). For \(\lambda\) in the weight lattice \(P\) of \(R\), the exponentials \(e^\lambda\) are the algebraic characters of a complex torus \(H\) of Lie algebra \(\mathfrak h\). Section 2 studies the action of the Dunkl-Cherednik operators \(T_\xi(k)\) (\(\xi \in \mathfrak h\)) on the \(\mathbb C\)-span \(\mathbb C[H]\) of the \(e^\lambda\) (\(\lambda \in P\)). A complete set \(\{E(\lambda, k):\lambda \in P\}\) of orthogonal polynomials which are common eigenfunctions of the \(T_\xi(k)\) is determined. Sections 3 and 4 are based on the work of Cherednik on Hecke algebras and their intertwiners. The intertwiners generate recursion relations between the eigenfunctions of the Dunkl-Cherednik operators with spectral parameters differing by an element of \(P\). Section 5 studies the shift principle. For each \(\lambda \in P\), the polynomials \(E(w\lambda,k)\) (\(w \in W\)) span a module \(M(\lambda,k)\) for the degenerate affine Hecke algebra. Each \(M(\lambda,k)\) contains (up to scalar multiples) a unique \(W\)-invariant element, the Jacobi polynomial \(P(\lambda,k)\). The shift operators relate Jacobi polynomials corresponding to different multiplicity parameters. The hypergeometric system of differential equations and its monodromy are presented in section 6. This leads to the definition of the hypergeometric function \(F(\lambda,k)\) of spectral parameter \(\lambda \in \mathfrak h^*\) associated with the root system \(R\). The \(E(k,\lambda)\) are the non-symmetric analogs of the Jacobi polynomials. Similarly, there are non-symmetric hypergeometric functions \(G(\lambda,k)\) associated with \(R\). The construction of the \(G(\lambda,k)\) leads naturally to the study of the Knizhnik-Zamalodchikov connection in section 7. The harmonic analysis on \(C_c^\infty(\mathfrak a)\) in the case of multiplicities \(k_\alpha \geq 0\) is studied in section 8, with the formulation of the Paley-Wiener theorem and the inversion and Plancherel formulas. By means of the residue calculus, the final section 9 extends the harmonic analysis to certain cases with \(k_\alpha<0\).

We now pass to the description of the second part of the memoirs. A complex reflection on a finite-dimensional Hilbert space \(V\) is a unitary transformation having a complex hyperplane as set of fixed points. A finite group \(W\) of unitary transformations of \(V\) is called a complex reflection group if it is generated by complex transformations. The irreducible complex transformation groups have been classified by G. C. Shephard and J. A. Todd [Can. J. Math 6, 274-304 (1954; Zbl 0055.14305)]. Let \(P\) denote the ring of polynomial functions on \(V\). By means of the natural action of \(W\) on \(P\) one can associate with each representation \(\tau\) of \(W\) a certain polynomial in one variables with nonnegative integral coefficients, called the fake degree of \(\tau\). M. Broué and G. Malle introduced in [Astérisque 212, 119-189 (1993; Zbl 0835.20064)] the cyclotomic Hecke algebra as a main tool for investigating the fake degrees. See also M. Broué, G. Malle and R. Rouquier [J. Reine Angew. Math. 500, 127-190 (1998; Zbl 0921.20046)], where also a “topological cyclotomic Hecke algebra” is introduced. In section 2 of the second part of these memoirs, an \(l\times l\) nonsingular matrix with entries in \(P\) is associated with every representation of \(W\) of finite dimension \(l\). This is the minimal \(\tau\)-matrix. For \(1\)-dimensional representations, the construction is due to R. P. Stanley [J. Algebra 49, 134-148 (1977; Zbl 0383.20029)]. In sections 3 and 4 the monodromy representation of the Knizhnik-Zamolodchikov connection is employed to construct the \(\tau\)-matrices. The main consequence is the fake degree symmetry proven in Theorem 4.2. The interpretation of these results in terms of the topological cyclotomic Hecke algebra is contained in section 6. The concluding section 7 presents several applications to the study of geometric Galois groups. Subsequent work of the author on Dunkl operators on complex reflection groups can be found in C. F. Dunkl and E. M. Opdam [Dunkl operators for complex reflection groups, Preprint NI01031-SFM, Isaac Newton Institute for Mathematical Sciences, Cambridge].

The theory of hypergeometric functions associated with root systems and Dunkl operators is one of the most important recent contribution to the harmonic analysis. Not only it has profoundly improved the understanding of the classical theory of Harish-Chandra’s spherical functions, but it has also provided tools and means which have been proven to be essential in both harmonic analysis and special function theory in more variables. These memoirs are therefore an indispensable source for all scientists with interests in these fields.

In the late 80ies, G.J. Heckman and E.M. Opdam defined the hypergeometric functions associated with root systems as analytic continuation of the Harish-Chandra’s spherical functions on Riemannian symmetric spaces. See G. J. Heckman and E. M. Opdam [Compos. Math. 64, 329-352 (1987; Zbl 0656.17006)], G. J. Heckman [ibid. 64, 353-373 (1987; Zbl 0656.17007)] and E. M. Opdam [ibid. 67, 21-49, 191-209 (1988; Zbl 0669.33007, Zbl 0669.33008)]. The analytically continued parameters are the root multiplicities. The definition of Heckman-Odpam’s hypergeometric functions is based on a generalization of the system of differential equations satisfied by Harish-Chandra’s spherical functions. The system represents the common spectral theory of a commuting family of differential operators, the so-called hypergeometric differential operators. In the symmetric case, they agree with the radial components on a maximal split Cartan subgroup of the invariant differential operators on the given Riemannian symmetric space. The construction of the hypergeometric differential operators became of completely algebraic nature when G. J. Heckman [Invent. Math. 103, 341-350 (1991; Zbl 0721.33009)] obtained them using a certain family of first-order non-commuting differential-reflection operators as building blocks. Heckman’s algebraic construction was later improved by I. Cherednik [Invent. Math. 106, No. 2, 411-432 (1991; Zbl 0742.20019)] with the introduction of another family of first-order differential-reflection operators which is moreover commutative. These operators are nowadays known as Dunkl-Cherednik operators. The harmonic analysis of the eigenfunctions of the Dunkl-Cherednik operators has been developed by E. M. Opdam in [Acta Math. 175, No. 1, 75-121 (1995; Zbl 0836.43017)].

To give a more detailed description of the first part of these lecture notes, let us fix some notation. Let \(\mathfrak a\) denote the Euclidean space with complexification \(\mathfrak h\). Let \(R \subset \mathfrak a^*\) be a root system with Weyl group \(W\), and let \(k=(k_\alpha)\) be a multiplicity function on \(R\). For \(\lambda\) in the weight lattice \(P\) of \(R\), the exponentials \(e^\lambda\) are the algebraic characters of a complex torus \(H\) of Lie algebra \(\mathfrak h\). Section 2 studies the action of the Dunkl-Cherednik operators \(T_\xi(k)\) (\(\xi \in \mathfrak h\)) on the \(\mathbb C\)-span \(\mathbb C[H]\) of the \(e^\lambda\) (\(\lambda \in P\)). A complete set \(\{E(\lambda, k):\lambda \in P\}\) of orthogonal polynomials which are common eigenfunctions of the \(T_\xi(k)\) is determined. Sections 3 and 4 are based on the work of Cherednik on Hecke algebras and their intertwiners. The intertwiners generate recursion relations between the eigenfunctions of the Dunkl-Cherednik operators with spectral parameters differing by an element of \(P\). Section 5 studies the shift principle. For each \(\lambda \in P\), the polynomials \(E(w\lambda,k)\) (\(w \in W\)) span a module \(M(\lambda,k)\) for the degenerate affine Hecke algebra. Each \(M(\lambda,k)\) contains (up to scalar multiples) a unique \(W\)-invariant element, the Jacobi polynomial \(P(\lambda,k)\). The shift operators relate Jacobi polynomials corresponding to different multiplicity parameters. The hypergeometric system of differential equations and its monodromy are presented in section 6. This leads to the definition of the hypergeometric function \(F(\lambda,k)\) of spectral parameter \(\lambda \in \mathfrak h^*\) associated with the root system \(R\). The \(E(k,\lambda)\) are the non-symmetric analogs of the Jacobi polynomials. Similarly, there are non-symmetric hypergeometric functions \(G(\lambda,k)\) associated with \(R\). The construction of the \(G(\lambda,k)\) leads naturally to the study of the Knizhnik-Zamalodchikov connection in section 7. The harmonic analysis on \(C_c^\infty(\mathfrak a)\) in the case of multiplicities \(k_\alpha \geq 0\) is studied in section 8, with the formulation of the Paley-Wiener theorem and the inversion and Plancherel formulas. By means of the residue calculus, the final section 9 extends the harmonic analysis to certain cases with \(k_\alpha<0\).

We now pass to the description of the second part of the memoirs. A complex reflection on a finite-dimensional Hilbert space \(V\) is a unitary transformation having a complex hyperplane as set of fixed points. A finite group \(W\) of unitary transformations of \(V\) is called a complex reflection group if it is generated by complex transformations. The irreducible complex transformation groups have been classified by G. C. Shephard and J. A. Todd [Can. J. Math 6, 274-304 (1954; Zbl 0055.14305)]. Let \(P\) denote the ring of polynomial functions on \(V\). By means of the natural action of \(W\) on \(P\) one can associate with each representation \(\tau\) of \(W\) a certain polynomial in one variables with nonnegative integral coefficients, called the fake degree of \(\tau\). M. Broué and G. Malle introduced in [Astérisque 212, 119-189 (1993; Zbl 0835.20064)] the cyclotomic Hecke algebra as a main tool for investigating the fake degrees. See also M. Broué, G. Malle and R. Rouquier [J. Reine Angew. Math. 500, 127-190 (1998; Zbl 0921.20046)], where also a “topological cyclotomic Hecke algebra” is introduced. In section 2 of the second part of these memoirs, an \(l\times l\) nonsingular matrix with entries in \(P\) is associated with every representation of \(W\) of finite dimension \(l\). This is the minimal \(\tau\)-matrix. For \(1\)-dimensional representations, the construction is due to R. P. Stanley [J. Algebra 49, 134-148 (1977; Zbl 0383.20029)]. In sections 3 and 4 the monodromy representation of the Knizhnik-Zamolodchikov connection is employed to construct the \(\tau\)-matrices. The main consequence is the fake degree symmetry proven in Theorem 4.2. The interpretation of these results in terms of the topological cyclotomic Hecke algebra is contained in section 6. The concluding section 7 presents several applications to the study of geometric Galois groups. Subsequent work of the author on Dunkl operators on complex reflection groups can be found in C. F. Dunkl and E. M. Opdam [Dunkl operators for complex reflection groups, Preprint NI01031-SFM, Isaac Newton Institute for Mathematical Sciences, Cambridge].

The theory of hypergeometric functions associated with root systems and Dunkl operators is one of the most important recent contribution to the harmonic analysis. Not only it has profoundly improved the understanding of the classical theory of Harish-Chandra’s spherical functions, but it has also provided tools and means which have been proven to be essential in both harmonic analysis and special function theory in more variables. These memoirs are therefore an indispensable source for all scientists with interests in these fields.

Reviewer: Angela Pasquale (Clausthal-Zellerfeld)

### MSC:

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

33C67 | Hypergeometric functions associated with root systems |

43A32 | Other transforms and operators of Fourier type |