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The answer at https://tex.stackexchange.com/a/151552/ notes that in pdfLaTeX the command\mathscr from the package mathrsfs only supports capital letters. Will some other package let me use \mathscr in lower case in pdfLaTeX?.

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    stix (and stix2) fonts (and packages of same name) have lowercase script alphabet
    – David Carlisle
    Mar 22, 2019 at 14:57

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You also can have lowercase script font with Adobe Bickham (bickham package) and with dutchcal:

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