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Fair Value Measurements - Schedule of Assets and Liabilities Measured at Fair Value on a Recurring Basis (Details) - USD ($) $ in Millions | Dec. 31, 2024 | Sep. 30, 2024
--- | --- | ---
Sponsored funds and separate accounts [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | $ 469.1 | $ 509.1
Investments related to long-term incentive plans [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 266.3 | 271.6
Other equity and debt investments [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 71.5 | 57.3
Fair Value, Recurring [Member] |  |
Assets [Abstract] |  |
Total Assets Measured at Fair Value | 806.9 | 838.0
Liability [Abstract] |  |
Securities sold short | 244.9 | 178.1
Contingent consideration liabilities | 27.5 | 28.2
Total Liabilities Measured at Fair Value | 272.4 | 206.3
Fair Value, Recurring [Member] | Level 1 [Member] |  |
Assets [Abstract] |  |
Total Assets Measured at Fair Value | 512.2 | 552.9
Liability [Abstract] |  |
Securities sold short | 244.9 | 178.1
Contingent consideration liabilities | 0.0 | 0.0
Total Liabilities Measured at Fair Value | 244.9 | 178.1
Fair Value, Recurring [Member] | Level 2 [Member] |  |
Assets [Abstract] |  |
Total Assets Measured at Fair Value | 185.3 | 168.5
Liability [Abstract] |  |
Securities sold short | 0.0 | 0.0
Contingent consideration liabilities | 0.0 | 0.0
Total Liabilities Measured at Fair Value | 0.0 | 0.0
Fair Value, Recurring [Member] | Level 3 [Member] |  |
Assets [Abstract] |  |
Total Assets Measured at Fair Value | 6.3 | 7.8
Liability [Abstract] |  |
Securities sold short | 0.0 | 0.0
Contingent consideration liabilities | 27.5 | 28.2
Total Liabilities Measured at Fair Value | 27.5 | 28.2
Fair Value, Recurring [Member] | NAV as a Practical Expedient [Member] |  |
Assets [Abstract] |  |
Total Assets Measured at Fair Value | 103.1 | 108.8
Liability [Abstract] |  |
Securities sold short | 0.0 | 0.0
Contingent consideration liabilities | 0.0 | 0.0
Total Liabilities Measured at Fair Value | 0.0 | 0.0
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 469.1 | 509.1
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 1 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 257.6 | 306.3
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 2 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 171.3 | 157.4
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 3 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 4.7 | 5.2
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | NAV as a Practical Expedient [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 35.5 | 40.2
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 266.3 | 271.6
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 1 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 237.1 | 242.5
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 2 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 0.0 | 0.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 3 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 0.0 | 0.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | NAV as a Practical Expedient [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 29.2 | 29.1
Fair Value, Recurring [Member] | Other equity and debt investments [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 71.5 | 57.3
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 1 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 17.5 | 4.1
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 2 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 14.0 | 11.1
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 3 [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | 1.6 | 2.6
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | NAV as a Practical Expedient [Member] |  |
Assets [Abstract] |  |
Investments, at fair value | $ 38.4 | $ 39.5

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