By Ivan Cherednik

This can be a detailed, basically self-contained, monograph in a brand new box of basic significance for illustration conception, Harmonic research, Mathematical Physics, and Combinatorics. it's a significant resource of basic information regarding the double affine Hecke algebra, also referred to as Cherednik's algebra, and its remarkable purposes. bankruptcy 1 is dedicated to the Knizhnik-Zamolodchikov equations hooked up to root structures and their family members to affine Hecke algebras, Kac-Moody algebras, and Fourier research. bankruptcy 2 incorporates a systematic exposition of the illustration conception of the one-dimensional DAHA. it's the easiest case yet faraway from trivial with deep connections within the idea of exact services. bankruptcy three is ready DAHA in complete generality, together with functions to Macdonald polynomials, Fourier transforms, Gauss-Selberg integrals, Verlinde algebras, and Gaussian sums. This e-book is designed for mathematicians and physicists, specialists and scholars, in case you are looking to grasp the hot double Hecke algebra method.

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Proof. Let m ∈ H(e). Then, we have for some u, u′ , v, v ′ ∈ M e = mu , m = eu′ , e = vm , m = v′ e . Therefore em = e(eu′ ) = eu′ = m and in the same way me = m. This shows that m ∈ eM e. Since m(eue) = mue = e , (eve)m = evm = e , 1032 1033 1034 the element m is both right and left invertible in M . Hence, m belongs to the group of units of eM e. Conversely, if m ∈ eM e is right and left invertible, we have mu = vm = e for some u, v ∈ eM e. Since m = em = me, we obtain mHe. J. Berstel, D. Perrin and C.

If P = Q, we say that it is a K-relation over Q. The set of all K-relations between P and Q is denoted by K P ×Q . Let m ∈ K P ×Q be a K-relation between P and Q. For p ∈ P , the row of index p of m is denoted by mp∗ . It is the element of K Q defined by (mp∗ )q = mpq . Similarly, the column of index q of m is denoted by m∗q . It is an element of K P . Let P, Q, R be three sets and let K be a complete semiring. For m ∈ K P ×Q and n ∈ K Q×R , the product mn is defined as the following element of K P ×R .

Proof. Set N (z) = I − M z, where I is the identity matrix and z is a variable. The polynomial N (z) can be considered both as a polynomial with coefficients in the ring of m × m-matrices or as an m × m-matrix with coefficients in the ring of real polynomials in the variable z. The polynomial N (z) is invertible in both structures, and its inverse N (z)−1 = (I − M z)−1 can in turn be viewed as a power series with coefficients in the ring of m × m-matrices or as a matrix whose coefficients are rational fractions in the variable z.

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