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Abstract Let G be a p -adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $$G=\mathrm {SO}(2n+1)$$ , $$\mathrm {Sp}(2n)$$ , and $$\mathrm {O}(2n)$$ .more » « less
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null (Ed.)We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $$\text{PGSp}_{6}$$ discovered by Pollack, we prove that a cuspidal representation of $$\text{PGSp}_{6}$$ is a (weak) functorial lift from the exceptional group $$G_{2}$$ if its (partial) Spin L-function has a pole at $s=1$ .more » « less
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