Download Elements of the Representation Theory of the Jacobi Group by Rolf Berndt PDF

By Rolf Berndt

After Pyatetski-Shapiro [PSI] and Satake [Sal] brought, self sustaining of each other, an early type of the Jacobi thought in 1969 (while no longer naming it as such), this idea was once given a distinct push by means of the e-book the speculation of Jacobi kinds through Eichler and Zagier in 1985. Now, there are a few assessment articles describing the advancements within the concept of the Jacobi crew and its automor phic types, for example via Skoruppa [Sk2], Berndt [Be5] and Kohnen [Ko]. We seek advice from those for extra old information and plenty of extra names of authors lively during this idea, which stretches now from quantity concept and algebraic geometry to theoretical physics. yet allow us to purely in brief point out a number of - occasionally very heavily comparable - issues touched by means of Jacobi concept as we see it: fields of meromorphic and rational features at the common elliptic curve resp. common abelian type constitution and projective embeddings of sure algebraic kinds and homogeneous areas correspondences among other kinds of modular types L-functions linked to other kinds of modular kinds and autom- phic representations precipitated representations invariant differential operators constitution of Hecke algebras decision of generalized Kac-Moody algebras and as a last target relating to the the following first pointed out combined Shimura types and combined explanations.

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L). 14) with 'Y = e- iO . 15) and N: = {(~ ~)(v,O,O) = n_(u,v): u,v E C}. Then we have the following central fact. 1 Remark. L* jd*.

Caution: Our character is precisely the inverse of the character defined in [Tate]. We have made our choice of characters analogous to that in the papers [Bel]-[Be6] in the real case. 2. The Schrodinger representation will be used throughout. 2 it is known that the map m ~ 'ljr identifies F with its own character group. It is also important to know that if F is discrete and" denotes the absolute different of F then is the greatest ideal of F, on which 'ljJ is trivial. In particular, if F = Qp, then 'ljJ is trivial on Zp and on no bigger ideal.

Hence the restriction of T to the matrix group must be irreducible. This group being isomorphic to R itself we see that our representation is one-dimensional and the matrices act through a unitary character of R. q, ---- C*. So the representations T from which we start our induction constitute a one-parameter family indexed by r E R. Putting everything together we have the following result. 2 Theorem. The irreducible unitary representations of GJ with trivial central character are exactly the following.

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