The conception requires the accepted angle of agency of automorphy j for Γ, which is a blazon of 1-cocycle in the accent of accumulation cohomology. The ethics of j may be circuitous numbers, or in actuality circuitous aboveboard matrices, agnate to the achievability of vector-valued automorphic forms. The cocycle action imposed on the agency of automorphy is article that can be commonly checked, back j is acquired from a Jacobian matrix, by agency of the alternation rule.
In the accepted setting, then, an automorphic anatomy is a action F on G (with ethics in some anchored finite-dimensional agent amplitude V, in the vector-valued case), accountable to three kinds of conditions:
to transform beneath adaptation by elements \gamma \in \Gamma according to the accustomed automorphy agency j;
to be an eigenfunction of assertive Casimir operators on G; and
to amuse some altitude on advance at infinity.
It is the aboriginal of these that makes F automorphic, that is, amuse an absorbing anatomic blueprint apropos F(g) with F(γg) for \gamma \in \Gamma . In the vector-valued case the blueprint can absorb a finite-dimensional accumulation representation ρ acting on the apparatus to 'twist' them. The Casimir abettor action says that some Laplacians accept F as eigenfunction; this ensures that F has accomplished analytic properties, but whether it is absolutely a complex-analytic action depends on the accurate case. The third action is to handle the case area G / Γ is not bunched but has cusps.
In the accepted setting, then, an automorphic anatomy is a action F on G (with ethics in some anchored finite-dimensional agent amplitude V, in the vector-valued case), accountable to three kinds of conditions:
to transform beneath adaptation by elements \gamma \in \Gamma according to the accustomed automorphy agency j;
to be an eigenfunction of assertive Casimir operators on G; and
to amuse some altitude on advance at infinity.
It is the aboriginal of these that makes F automorphic, that is, amuse an absorbing anatomic blueprint apropos F(g) with F(γg) for \gamma \in \Gamma . In the vector-valued case the blueprint can absorb a finite-dimensional accumulation representation ρ acting on the apparatus to 'twist' them. The Casimir abettor action says that some Laplacians accept F as eigenfunction; this ensures that F has accomplished analytic properties, but whether it is absolutely a complex-analytic action depends on the accurate case. The third action is to handle the case area G / Γ is not bunched but has cusps.
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