The Gel’fand realization and the exceptional representations of SL(2,R)

1985 ◽  
Vol 26 (1) ◽  
pp. 12-17 ◽  
Author(s):  
Debabrata Basu ◽  
T. Bhattacharya
Keyword(s):  
Exceptional Representations

scholarly journals On spinor exceptional representations

1982 ◽  
Vol 87 ◽  
pp. 247-260 ◽  
Author(s):  
J. W. Benham ◽  
J. S. Hsia
Keyword(s):  
Quadratic Form ◽  
Real Numbers ◽  
The Real ◽  
Integer Coefficients ◽  
Spinor Genus ◽  
Exceptional Representations

Let f(x1 …, xm) be a quadratic form with integer coefficients and c ∈ Z. If f(x) = c has a solution over the real numbers and if f(x) ≡ c (mod N) is soluble for every modulus N, then at least some form h in the genus of f represents c. If m ≧ 4 one may further conclude that h belongs to the spinor genus of f. This does not hold when m = 3.

scholarly journals THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

2015 ◽  
Vol 16 (3) ◽  
pp. 609-671 ◽  
Author(s):  
Eyal Kaplan
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Automorphic Representation ◽  
Double Cover ◽  
Spin Groups ◽  
Principal Series Representation ◽  
Double Covers ◽  
Exceptional Representations ◽  
Period Integral ◽  
Exceptional Character

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of$\mathit{SO}_{2n+1}$of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

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