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On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras. (English) Zbl 1173.11050
Let $$\mathcal D$$ be a non-split simple algebra of dimension $$m = 4$$ or $$9$$ over a number field $$k$$, and define a prehomogeneous vector space $$(G,V) = (\mathcal D^\times \times (\mathcal D^{op})^\times \times \mathrm{GL}(2), \mathcal D \otimes k^2)$$ with action $$(g_1, g_2, g)\cdot (a\otimes v) = g_1ag_2 \otimes gv$$, which is an inner $$k$$-form of $$(\mathrm{GL}(n) \times \mathrm{GL}(n) \times \mathrm{GL}(2), k^n \otimes k^n \otimes k^2)$$. Then, one may define the global zeta function (Shintani zeta function) $$Z(\Phi, s, \omega)$$ associated with the above $$(G,V)$$, where $$\Phi$$ is a Schwartz-Bruhat function on $$V_\mathbb A$$, $$s \in \mathbb C$$ and $$\omega$$ is an adelic character of $$G$$. The main result of this paper is the meromorphic continuation of $$Z(\Phi, s, \omega)$$ to the region $$\text{Re}(s) > 2m -2$$ with an only possible pole at $$s = 2m$$.
For the case $$m = 4$$, the author obtained a certain density theorem as an application in a later paper [ibid. 58, No. 2, 625–670 (2008; Zbl 1231.11111)].
There are preceding works for the split case and another non-split case by A. C. Kable, D. J. Wright and A. Yukie.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11S90 Prehomogeneous vector spaces
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##### References:
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