New PDF release: Automorphic Forms on Semisimple Lie Groups

By Bhartendu Harishchandra, J.G.M. Mars

ISBN-10: 3540042326

ISBN-13: 9783540042327

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45) This is clearly a linear character of Z„, because X ( r ) (zs)X ( r ) (z`) = Es+r=X(r)(Zs+t). 45). As r ranges from 0 to n —1, we obtain n characters, as required. Contrary to our usual convention the trivial character will here be denoted by X (0) . The complete character table of Z,, is then summarised by the equations X (r ) (zs) = e(2,rirs/n) (r, s = 0, 1, ... , n — 1). ,n-1) could be checked; they follow from elementary identities for exponential functions. Example. The group Z3 consists of the elements 1, z, z 2 (z 3 = 1).

Ag x8 where a l , a 2 , ... , ag are arbitrary complex numbers. Let w=Nlx1 +fl2x2+ ... + flaxa be another element of G. Then y = w if and only if a ; = /3 (i = 1, 2, ... 20) 1v=v. Furthermore, G c possesses a multiplicative structure. 21) where j (i, j) is a well-defined integer lying between 1 and g. i)• This multiplication is associative by virtue of the associative law for G. Thus, if u, y, w E G e , then (uv)w = u(vw). As we have already remarked (p. 27), a vector space which is endowed with an associative multiplication is called an algebra.

Exercises 1. Let H be a subgroup of G with coset representatives t„ t2 , ... , t„. Show that the kernel of the permutation representation (p. 2) Ht l Ht 2 Ht„ 1 Ht,x Ht 2x . Ht„x1 is the group „ n t 1Ht . ; ; ^-^ 2. Show that the map a^C(a)—^ -0 1 -1/ 1 defines a representation of the cyclic group gp{a I =1 }. Prove that this representation is irreducible over the field of real numbers. 3. Let E be a linear map of the m-dimensional vector space V into itself such that E2 = E, e # i (the identity map).

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Automorphic Forms on Semisimple Lie Groups by Bhartendu Harishchandra, J.G.M. Mars


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