By Heckman G., Schlichtkrull H.

This sharply concentrated ebook treats the massive classification of hypergeometric features utilized in the answer of differential equations and in physics. even if identified from the time of Euler, those services can now be understood for the 1st time within the context of harmonic research and symmetric areas. Divided into elements, the cloth in Harmonic research and targeted features on Symmetric areas is predicated on lectures given for the "European college of staff Theory," a sophisticated path on present advancements in staff conception. The authors supply scholars and researchers with an intensive and considerate evaluation, elaborating at the subject with transparent statements of definitions and theorems and augmenting those with time-saving examples. an intensive set of notes supplementations the textual content. The publication leads readers from the basics of semisimple symmetric areas to the Reimannian case. The nineteenth century paintings of Euler, Gauss, Kummer, Riemann, and Klein on hypergeometric features is associated with root structures and symmetric areas. Algebraic and analytic equipment are used, with many connections made within the geometric context of symmetric areas. This quantity will curiosity harmonic analysts, these engaged on or utilising the idea of symmetric areas, and people with an curiosity in distinctive services.

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**Example text**

2) defines a partial compacti- and using the action of the Weyl group W this can be extended to a smooth global compactification of H / C . This is nothing but the toroidal compactification corresponding to the decomposition of a into Weyl chambers (see for example [11, 55]). 2) onto (0, 1) n. 1. 2) has the picture X2 Xl=l x2=l X1X2 = r 1 X1 The point (1, 1) is the image of the identity element, and the curves X 2 - - 1, X l X 2 - - Xl -- 1, 1 are the image of {h E H; A(h) -- 0}. Let { r ] l , .

12) Observe that with our choice of d(A, k) we have {d(~,k)r~(~,k)}~:~o - 0 Vn <_ 0 with n / ; n0. Hence {d(A,k)O()~,k;a)}~:~ o is a multiple of O(,k0+n0, k;a). 1) of hypergeometric differential equations. 1) with leading exponent wA - p(k), w C W. 5. 14) (A, c~v) - j for some (~ E R+, some j E N - { 1 , 2 , . . }. Proof. The fact that for certain A E ~* (cf. 7)) the recurrence rela- tions break down is the phenomenon of logarithmic terms caused by the differences of exponents being integers. 15) , k - wA C Q for some w E W, w 7~ 1.

1). 7. 1) then U---(OqlU,... 12). Conversely, if U - ( U l , . . 1) and uj - OqjUl. Proof. , P ( u ) - 0 V P e J()~, k). If we write U - (Oql u, . . 9) that ( P + A ( P ) ) ( U ) = 0 VP E ffl | Ui?. 12). Now suppose U - ( u l , . . 12). 11) and induction on the order of differential operators it is easy to see that ( P + A ( P ) ) ( U ) - 0 V P E 9~ | UO. Since A l j ( P ) - 0 for P E J(~, k) we get P ( u l ) - 0 V P e J()~, k). Moreover uj - Oaf(U1) because Alj(Oq~) - - b i j . 8. 1) has dimension d = [W I and consists of holomorphic functions.