# Equidimensional adic eigenvarities for groups with discrete series

Gulotta, D

9 October 2019

## Journal:

Algebra and Number Theory

## Last Updated:

2021-07-20T05:06:48.5+01:00

8

13

## DOI:

10.2140/ant.2019.13.1907

1907-1940

## abstract:

<p style="text-align:justify;"> We extend Urban’s construction of eigenvarieties for reductive groups <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi>[/itex] such that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℝ</mi></mrow><mo class="MathClass-close">)</mo></mrow>[/itex] has discrete series to include characteristic <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi>[/itex] points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi>[/itex] -analytic manifold taking values in a complete Tate <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>p</mi></mrow></msub>[/itex] -algebra in which <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi>[/itex] is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi>[/itex] -adic Lie groups given by Johansson and Newton. </p>

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