Timeline for On the reductive group
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2014 at 23:20 | comment | added | Monty | Recently, I found that why the reductive group condition should be imposed in automorphic form theory. As Paul said, we can define automorphic representation for arbitrary algebraic group. However, to define its $L$-function, we should consider only reductive group, because the Langlands dual group can be defined only for reductive and one ingredient to define the local $L$-function is representation of this Langlands dual group. | |
| Oct 6, 2014 at 23:04 | vote | accept | Monty | ||
| Oct 3, 2014 at 12:27 | comment | added | Monty | @YCor, Thank you for your penetrating example. | |
| Oct 2, 2014 at 9:35 | comment | added | YCor | $SU(n)\times G_a$ as the same Lie algebra as the unitary group $U(n)$ but is not reductive. | |
| Oct 2, 2014 at 5:59 | comment | added | Monty | @user52824, Thanks for your illumination. The geometric property of unitary group is the key to be reductive. | |
| Oct 2, 2014 at 5:55 | vote | accept | Monty | ||
| Oct 6, 2014 at 23:04 | |||||
| Oct 2, 2014 at 1:27 | comment | added | user27920 | As Paul Garrett knows, unitary groups can be defined rather more generally for certain hermitian forms relative to a separable quadratic extension of any field at all (even in char > 0, where Lie algebras are insufficient). Reductivity is a property of an algebraic group, rather than a Lie-group concept (though it is not so distant from Lie theory), and as such it is a geometric property, testable over an algebraic closure of the ground field (unlike pseudo-reductivity, for example). Since a unitary group becomes a general linear group after scalar extension to an algebraic closure... | |
| Oct 2, 2014 at 0:56 | comment | added | Monty | @garret, Thanks for you kind comment. Would you please let some heuristic example that is not reductive and what property hinders it from being reductive? Since the definition of the reductive group touch not my heart and so if you suggest one one couterexample, that would help me. | |
| Oct 1, 2014 at 23:58 | history | answered | paul garrett | CC BY-SA 3.0 |