feat(RepresentationTheory): add Smooth.Basic for smooth representations

[JX-Mo]

This PR introduces basic set up for smooth representations of a topological group. Hopefully it will serve as preliminary content for those who are interested in formalizing representation theory for p-adic reductive groups in the future. We introduce the following in the file:

1. Define smoothness for representations of a topological group. It is a property for a representation to be smooth: the stabilizer of any vector is open.
2. Prove basic closure properties: subrepresentations, quotient representations, tensor products, direct sums of smooth representations are smooth.
3. Construct smoothHom and contragredient: they are obtained by cutting out smooth vectors from the naive hom and dual.

• depends on feat(RepresentationTheory): add stabilizers of vectors in representations #40941
• depends on feat(RepresentationTheory): add Subrepresentation.toRepresentation_apply and Subrepresentation.quotient #41081

[github-actions]

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[github-actions]

PR summary 448cad44a8

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RepresentationTheory.Stabilizer (new file)	1493
Mathlib.RepresentationTheory.Smooth.Basic (new file)	1620

---

Declarations diff (regex)

+ IntertwiningMap.isSmoothVector
+ IntertwiningMap.isSmooth_injective
+ IntertwiningMap.isSmooth_surjective
+ IntertwiningMap.smoothVectors
+ IntertwiningMap.stabilizer_le
+ IsSmooth
+ IsSmoothVector
+ contragredient
+ isSmoothVector_add
+ isSmoothVector_apply
+ isSmoothVector_iff
+ isSmoothVector_smul
+ isSmoothVector_sum
+ isSmoothVector_tmul
+ isSmoothVector_zero
+ isSmooth_contragredient
+ isSmooth_directSum
+ isSmooth_iff
+ isSmooth_prod
+ isSmooth_quotient
+ isSmooth_smoothHom
+ isSmooth_smoothVectors
+ isSmooth_subrepresentation
+ isSmooth_tprod
+ isSmooth_trivial
+ le_stabilizer_add
+ le_stabilizer_smul
+ le_stabilizer_sum
+ le_stabilizer_tmul
+ mem_smoothSubmodule
+ mem_stabilizer
+ quotient
+ quotient_apply_mk
+ smoothHom
+ smoothSubmodule
+ smoothVectors
+ stabilizer
+ stabilizer_conj
+ stabilizer_zero
+ toRepresentation_apply_coe
+ toRepresentation_apply_mk

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 448cad4).

• +47 new declarations
• −0 removed declarations

+Representation.IntertwiningMap.stabilizer_le
+Representation.Smooth.IntertwiningMap.isSmoothVector
+Representation.Smooth.IntertwiningMap.isSmooth_injective
+Representation.Smooth.IntertwiningMap.isSmooth_surjective
+Representation.Smooth.IntertwiningMap.smoothVectors
+Representation.Smooth.IsSmooth
+Representation.Smooth.IsSmooth.casesOn
+Representation.Smooth.IsSmooth.mk
+Representation.Smooth.IsSmooth.rec
+Representation.Smooth.IsSmooth.recOn
+Representation.Smooth.IsSmooth.smooth
+Representation.Smooth.IsSmoothVector
+Representation.Smooth.contragredient
+Representation.Smooth.isSmoothVector_add
+Representation.Smooth.isSmoothVector_apply
+Representation.Smooth.isSmoothVector_iff
+Representation.Smooth.isSmoothVector_smul
+Representation.Smooth.isSmoothVector_sum
+Representation.Smooth.isSmoothVector_tmul
+Representation.Smooth.isSmoothVector_zero
+Representation.Smooth.isSmooth_contragredient
+Representation.Smooth.isSmooth_directSum
+Representation.Smooth.isSmooth_iff
+Representation.Smooth.isSmooth_prod
+Representation.Smooth.isSmooth_quotient
+Representation.Smooth.isSmooth_smoothHom
+Representation.Smooth.isSmooth_smoothVectors
+Representation.Smooth.isSmooth_subrepresentation
+Representation.Smooth.isSmooth_tprod
+Representation.Smooth.isSmooth_trivial
+Representation.Smooth.mem_smoothSubmodule
+Representation.Smooth.smoothHom
+Representation.Smooth.smoothSubmodule
+Representation.Smooth.smoothVectors
+Representation.Smooth.smoothVectors.congr_simp
+Representation.le_stabilizer_add
+Representation.le_stabilizer_smul
+Representation.le_stabilizer_sum
+Representation.le_stabilizer_tmul
+Representation.mem_stabilizer
+Representation.stabilizer
+Representation.stabilizer_conj
+Representation.stabilizer_zero
+Subrepresentation.quotient
+Subrepresentation.quotient_apply_mk
+Subrepresentation.toRepresentation_apply_coe
+Subrepresentation.toRepresentation_apply_mk

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type (weak)
5010	2	exposed public sections

Current commit 448cad44a8
Reference commit 46931f419b

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[JX-Mo]

Could someone please add the t-algebra label?

[felixpernegger]

Could someone please add the t-algebra label?

For the future, note you can write "t-algebra" as comment and then the label will be applied.