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LSpice
LSpice
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A lattice in $ SL_n\operatorname{SL}_n $ is Ad-irreducible

Let$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ SL_n $$ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?

For example, I think it is true that all lattices in $ SL_2(\mathbb{R}) $$ \SL_2(\mathbb{R}) $ and $ SL_2(\mathbb{C}) $$ \SL_2(\mathbb{C}) $ are Ad-irreducible.

A lattice in $ SL_n $ is Ad-irreducible

Let $ G $ be a noncompact simple Lie group. For example $ SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?

For example, I think it is true that all lattices in $ SL_2(\mathbb{R}) $ and $ SL_2(\mathbb{C}) $ are Ad-irreducible.

A lattice in $ \operatorname{SL}_n $ is Ad-irreducible

$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?

For example, I think it is true that all lattices in $ \SL_2(\mathbb{R}) $ and $ \SL_2(\mathbb{C}) $ are Ad-irreducible.

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Ian Gershon Teixeira
Ian Gershon Teixeira
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A lattice in $ SL_n $ is Ad-irreducible

Let $ G $ be a noncompact simple Lie group. For example $ SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?

For example, I think it is true that all lattices in $ SL_2(\mathbb{R}) $ and $ SL_2(\mathbb{C}) $ are Ad-irreducible.